President Volodymyr Zelensky of Ukraine has called for a new peace summit in November, one open to participation by Russia. American and European officials privately wonder whether the Ukrainian and Russian militaries are so exhausted that they will pause fighting. And Western diplomats are discussing what kind of defense alliance their nations could promise Ukraine to help reach a settlement, officials say.

That kind of background chatter would seem to indicate a greater willingness by Ukraine and its allies to eventually engage in peace talks with Russia. But the most urgent diplomacy taking place now — and discussed with intensity this week in Kyiv, Washington and London — is about shaping the battlefield in Ukraine’s favor.

Ukrainian and Western officials say President Vladimir V. Putin still shows no signs of being willing to engage in peace talks in good faith. American and European officials, speaking on condition of anonymity to discuss sensitive diplomacy, have made clear over the last week that they are operating on the idea that Ukraine must consolidate and expand its gains on the battlefield in order to prod Mr. Putin toward the negotiating table — and to have meaningful leverage if talks start.

The discussions have been energized by the Ukrainian military’s occupation of Russian territory in Kursk, an assault that surprised both Russia and Ukraine’s partners after it began last month.

Ukraine has also intensified strikes against some Russian oil production facilities, hitting at the lifeblood of its enemy’s economy. And it has largely neutralized Russia’s Black Sea naval fleet.

Diplomatic discussions this week have focused on whether the United States, Britain and France should give Ukraine permission to use long-range missiles they have provided to strike deep into Russia. Right now, those countries only allow Ukraine to hit military targets just over the Russian border.

Britain and France are ready to permit longer strikes with their own weapons, but are waiting for President Biden to sign off, which he is poised to do, U.S. officials say. Mr. Biden remains hesitant to allow Ukraine to use American weapons for deep strikes, say the officials.

Mr. Biden and Keir Starmer, the British prime minister, plan to discuss the issue when they meet in Washington on Friday.

In Kyiv on Wednesday, Mr. Zelensky urged Secretary of State Antony J. Blinken and David Lammy, the British foreign secretary, to loosen restrictions on the weapons. And Mr. Blinken said afterward at news conferences in both Kyiv and Warsaw, Poland, that the United States would “adjust and adapt” to conditions on the battlefield.

“We’re determined to ensure that they have what they need to succeed,” Mr. Blinken told reporters in Kyiv.

Battlefield success underpins any hopes for a negotiated settlement, said Tymofiy Mylovanov, the president of the Kyiv School of Economics and a former minister of economic development. It is precisely this kind of enhanced military capability, he said in an interview, that could bring Mr. Putin to the negotiating table and help sustain any truce that might result, especially since economic sanctions have failed to change his behavior.

To have leverage, he said, Ukraine needs to be able to quickly strike ports, airfields and oil facilities — and even threaten Moscow with missile attacks, if necessary.

“Putin doesn’t believe in any diplomacy; he’s just playing this game,” Mr. Mylovanov said. “But I think he also wants to see evidence. You know, bring the receipts. So, does Ukraine have the capabilities?”

Russia’s goals also depend on fulfilling battlefield needs, and its diplomatic discussions with partners revolve around that. That was underscored this week by the public accusation by Mr. Blinken and Mr. Lammy that Iran had begun shipping short-range ballistic missiles to Russia.

Mr. Blinken said in London that Russia would use the missiles to hit Ukraine “within weeks.” Though Iran and Russia have denied the existence of the shipments, what has been clear throughout the war is that Russia’s diplomacy has been focused on bolstering the Russian military, mainly with help from China, North Korea and Iran.

For years, Mr. Biden, fearing escalation toward a Russia-versus-NATO war, has acquiesced to many of Mr. Zelensky’s specific arms requests only after much negotiation and internal hand-wringing.

Critics of Mr. Biden’s gradual approach say that it ultimately undermines long-term diplomacy in a couple of ways: first, by contributing to an atmosphere of frustration and distrust between American and Ukrainian officials; and second, by not giving the Ukrainians the capabilities they need to accrue bargaining chips that would force Mr. Putin to negotiate.

In effect, they say, Mr. Biden and his aides have allowed Mr. Putin to shape the war in his favor by instilling in Washington a fear of crossing unknowable “red lines” in the Russian leader’s head.

“We were suddenly in a war where a lot of us are convinced that it’s normal for Russians to be in Ukraine, but it’s not normal for Ukrainians to be in Russia, which is a total innovation in the history of warfare,” said Timothy Snyder, a Yale historian who was in Ukraine this week as a representative for United24, a Ukrainian state fund-raising group. “Nobody has ever claimed that before, let alone made their enemies believe it.”

“That is an incredible psychological success on the Russian side,” he said. “And there was no way for the Ukrainians to get us to think any way else except by proving it, which is what Kursk did, right?”

The Ukrainian push into Kursk has slowed as Russian forces counterattack, but if Kyiv can maintain its footprint there and maybe even seize more Russian territory, then it would have a stronger hand in any future peace talks.

William H. McRaven, a retired U.S. admiral who was a commander of special operations forces in the Obama administration, said that until Kursk, he had thought the best Ukraine could hope for was an armistice, security guarantees from allies, and money from frozen Russian assets for rebuilding. But the Kursk invasion surprised him, and it could give Mr. Zelensky a new card to play, he said.

“It boosts the morale in Ukraine right now,” he said last week at the Texas Tribune Festival in Austin. “Morale is always important, but probably more important now than ever before.”

The shape of diplomacy will also pivot on the U.S. presidential election in November. Vice President Kamala Harris has vowed to continue the Biden administration’s support of Ukraine. Former president Donald J. Trump has led many Republican lawmakers in trying to block aid to Ukraine.

When asked by a debate moderator on Tuesday whether he wanted Ukraine to win, Mr. Trump avoided saying yes.

Mr. Trump has said he could end the war, without giving details. He might see halting military aid to Ukraine as one way of forcing Mr. Zelensky to the table with Mr. Putin — at a severe disadvantage.

In an interview with “The Shawn Ryan Show” that was posted on Thursday, Mr. Trump’s vice-presidential nominee, JD Vance, laid out his vision of a peace settlement for Ukraine. It aligns with Mr. Putin’s desires: Ukraine cedes territory that the Russian military occupies; the two nations remain separated by a demilitarized zone; and Ukraine promises not to join NATO or other “allied institutions.”

Mr. Zelensky has been more vocal recently on the possibility of an eventual settlement, but only on terms that he and most Ukrainian citizens can accept. Ukraine regards Mr. Putin’s stated conditions as a demand for surrender.

The Ukrainian president has said he will present Mr. Biden, Ms. Harris and Mr. Trump this month with a “victory plan” that would “influence Russia’s decision to end this war.” The strategy would involve putting psychological, political, diplomatic and — most immediately — military pressure on Moscow. Mr. Zelensky also plans to speak at an international peace summit in November, the second one this year.

This is a war of attrition, and ultimately the key to peace talks is pounding home to Mr. Putin that Ukraine and its allies are committed to and capable of waging a long-term military campaign, said Oleksandr Merezhko, chairman of the foreign affairs committee of Ukraine’s Parliament.

“When he sees that Ukrainian society is not divided, is determined to survive and hopefully to win, the West is determined to provide all the necessary equipment, that we can target Russian military targets deep into Russian territory, then he will start to think about something,” Mr. Merezhko said.

Russia announced on Friday that it had decided to expel six British diplomats from the country, accusing them of engaging in espionage and sabotage work, in a move that highlighted the deepening tensions between Moscow and London.

The Russian Federal Security Service, or F.SB., said that the decision had been made in response to “the numerous unfriendly steps taken by London,” an apparent reference to signals from Britain that it is eager to allow Ukraine to use its “Storm Shadow” long-range missiles against military targets deep inside Russia.

Speaking on Thursday about that potential shift on the use of missiles, President Vladimir V. Putin warned that such a decision would mean that NATO countries were “at war with Russia” and that it would “clearly change the very essence, the very nature of the conflict.”

On Friday, the Kremlin’s spokesman, Dmitri S. Peskov, told reporters that Mr. Putin’s statement was “very important” and that the Russian government had “no doubt” that the message “had reached its addressees.”

The British Foreign Office — which said the accusations of spying by the diplomats were baseless — said that the six had left Russia last month, after Moscow’s notification to London about its decision in early August.

That came after Britain said in May that it was expelling a Russian defense attaché as an undeclared military intelligence officer and that it was imposing other restrictions on the Russian Embassy in London.

At the time, Moscow responded by ordering the British defense attaché to leave Russia. The tit-for-tat exchanges have come amid the backdrop of the war in Ukraine, and the timing of the latest Russian announcement coincided with Prime Minister Keir Starmer’s trip to Washington on Friday.

The proposed use of Britain’s “Storm Shadow” missiles — which would allow the Ukrainian military to strike in Russian territory far from the border — is high on Mr. Starmer’s agenda for the meeting. Britain wants explicit permission from President Biden in order to demonstrate a coordinated strategy with the United States and France, which has virtually the same model of missile.

The F.S.B., the main successor agency to the Soviet-era K.G.B., said that the six expelled diplomats had been sent to Russia by the British foreign service directorate responsible for Eastern Europe and Central Asia.

Since the invasion of Ukraine, the agency’s statement said, the British directorate has turned into “a special service, whose main task is to inflict strategic defeat” on Russia.

The F.S.B. said that the activities of the six diplomats were “threatening” to Russia’s security and that “signs of spying and sabotage” had been detected in their work.

In a statement, the British Foreign Office said, “The Russian authorities revoked the diplomatic accreditation of six U.K. diplomats in Russia last month, following action taken by the U.K. government in response to Russian state-directed activity across Europe and in the U.K.” It added, “We are unapologetic about protecting our national interests.”

Maria V. Zakharova, a spokeswoman for the Russian Foreign Ministry, said in a statement that her ministry supported the F.S.B.’s decision.

She said that the six British diplomats had been engaged in “subversive work aimed at harming” Russian people. In a separate statement, the Russian Foreign Ministry promised to expel more British diplomats should they engage in similar activities.

Mr. Peskov, the Kremlin spokesman, said on Friday that there was “no talk” about a complete closure of the British Embassy in the Russian capital.

Russian state television showed pictures of the six diplomats and identified them, although the names have not been confirmed. The diplomats had met with Russian civil society activists, the TV report said.

In 2018, Russia expelled 23 British diplomats and shut down the British Council, an organization devoted to international cultural and educational opportunities, in the country after London sent home the same number of Russian diplomats. That came amid a diplomatic crisis touched off by a nerve agent attack in Britain against a Russian former intelligence agent.

While Moscow and London have shared hostility toward each other for years, the relationship worsened after the Russian invasion of Ukraine in 2022. Britain is habitually portrayed by Russian state television as among the most aggressive and Russophobic states in the West.

Stephen Castle contributed reporting from London.

The border between Egypt and Gaza has become a major point of contention in the negotiations over a cease-fire to end the war in Gaza — not just between Israel and Hamas, but also between Israel and Egypt.

Israel’s prime minister, Benjamin Netanyahu, says the country must occupy the border area to prevent Hamas from smuggling weapons into the Gaza Strip from Egypt’s Sinai Peninsula. He has cast control of the Philadelphi Corridor, as the border zone is known, as a matter of existential importance for Israel, though some Israeli politicians believe he is using the issue to avoid reaching a deal for a cease-fire and the release of hostages.

Egypt argues that it has already cracked down on smuggling there — because doing so serves its own security interests, not just Israel’s — by building barriers, destroying tunnels and patrolling the area. For Cairo, accepting Israeli troops on the border would threaten its own national security and draw criticism from the Egyptian public, analysts say.

The increasingly bitter dispute has destabilized the once strong security partnership between Egypt and Israel, whose landmark 1979 peace treaty has been a linchpin of Middle East geopolitics for decades.

Here’s a look at the issue.

What has Hamas been able to smuggle through the border?

Exactly how much Hamas has been able to smuggle into Gaza via the Gaza-Egypt border — above ground or through tunnels — is unknown.

The question has come under fierce debate even in Israel: Some Israelis — who want to see an agreement reached to free Israeli hostages — have played down the issue of smuggling through the Philadelphi Corridor. But others who want Israel to keep fighting to destroy Hamas have emphasized the danger to Israel’s security.

That, along with the difficulty of obtaining independent information, makes it difficult to assess the scale of the smuggling.

Current and former Israeli security officials have often said that tunnels and insufficient screening at the Rafah crossing between Egypt and Israel have allowed Hamas to stock up on weaponry and other supplies that can be turned into weapons.

Hamas has smuggled in components of long-range rockets, rocket-propelled grenades, sniper rifles, machine guns, bullets and other weapons, according to an Israeli security official, who spoke on the condition of anonymity because he was not authorized to communicate with the media.

Nadav Argaman, a former director of Israel’s Shin Bet security agency, said in an interview with Israel’s Channel 12 that only “a very small amount of weapons” had been smuggled into Gaza through tunnels since Egypt acted to clear the border zone of tunnels and infrastructure. Most smuggling had instead occurred aboveground at the Rafah crossing, he said, adding that there was “no connection between the weapons in Gaza and the Philadelphi Corridor.”

Hamas has also heavily relied on other routes, including small fishing boats coming by sea and through the Kerem Shalom crossing between southern Gaza and Israel, three Israeli defense officials said.

And Hamas has failed to import sophisticated anti-tank missiles, a sign that Egypt’s efforts to stamp out smuggling have had success, two of the three defense officials said. Most of Hamas’s weapons have been locally made, the officials said. In contrast, Hezbollah, a Lebanese militant group and a beneficiary of Iranian support, like Hamas, has repeatedly fired advanced anti-tank missiles against Israel.

Still Hamas has publicly acknowledged receiving arms from abroad. In May 2021, Yahya Sinwar, then Hamas’s political leader in Gaza, singled out Iran for praise, thanking it for not being “stingy” in supplying the group with weapons.

Eyal Hulata, a former Israeli national security adviser, agreed that much of the smuggling occurred above ground through the Rafah crossing, which before the war served as a conduit for people to move between Egypt and Gaza. Still, in an interview he described underground smuggling in the area as a “serious problem.”

A dossier compiled by an Arab intelligence agency in April and obtained by The Times says that some 2,500 tunnels once existed between Gaza and Egypt, but that most were destroyed in an Egyptian government crackdown between 2013 and 2016.

Around 10 tunnels existed as of early 2024, but increased Egyptian oversight has eroded smuggling in recent years, the document says. Two Israeli defense officials also said Egypt had stepped up anti-smuggling enforcement in recent years, especially regarding the tunnels.

But in their heyday, the tunnels frustrated Israeli officials because they allowed for both weapons and goods like fuel to enter Gaza unchecked.

Egypt considers Hamas a security threat and is eager to prevent it from arming itself. Yet, the document adds, Egyptian personnel at the border often accept bribes to look the other way.

What does Israel say about smuggling there?

In a news conference on Wednesday evening, Mr. Netanyahu insisted that Israel would not agree to a cease-fire unless Israel could secure the corridor.

“Somebody has to be there,” he said, adding that until “they can actually prevent the recurrence of what happened there before,” Israel should remain.

Control of the Philadelphi corridor only appeared as a serious stumbling block in the past few weeks. Before that, Mr. Netanyahu barely mentioned it in public, if at all, leading some of his critics to say he was using the matter as a way to sabotage talks.

Israel, with U.S. backing, has previously pressed Egypt to heighten security along the border by building a high wall extending below ground, with sensor systems that would alert both the Israeli and Egyptian militaries to tunneling and smuggling, according to Egyptian and Israeli security experts.

Such a system would be similar to the 40-mile underground wall, outfitted with hundreds of cameras, radars and sensors, that Israel built after uncovering tunnels from Gaza into Israel.

Some have called for advanced screening equipment to be installed at the Rafah crossing to prevent smuggling there.

What does Egypt say?

Egypt says it has effectively cut off smuggling routes into Gaza. Years ago, it destroyed the main smuggling tunnels, flooded them with seawater and razed the buildings that provided cover for people using the tunnels. Egypt also destroyed its side of Rafah, a city that formerly straddled both sides of the Egypt-Gaza border, and moved Egyptian residents out.

Egypt views Hamas, an offshoot of an Egyptian Islamist group that Cairo considers an enemy, as an existential threat as well.

“Netanyahu’s attempt to blame Egypt for all the smuggling is total, utter nonsense,” said Abdel Monem Said Aly, a pro-government Egyptian political analyst.

Egypt believes the presence of Israeli troops in the corridor risks further angering the Egyptian public. And Egypt wants to show that it can manage the border itself.

Egypt is also extremely sensitive to the danger that Israel might use its control of the border to push Gazans to flee into Sinai.

What does it mean for the relationship?

Earlier this year, Egypt warned Israel that its actions at Rafah and in the border zone could constitute a violation of the two countries’ peace treaty. In recent weeks, it has accused Israel of fixating on the border area to stall a cease-fire.

Following Mr. Netanyahu’s Wednesday remarks, Al Qahera News, a state-owned broadcaster, quoted an unnamed senior official accusing Israel of overlooking other smuggling routes and blaming Egypt to cover up Israel’s own “failures.”

Yet neither side appears willing to give up on the peace treaty. Israel has tried to win more Arab partners, not enemies. For Egypt, the accord has generated valuable military and intelligence cooperation against an insurgency in the Sinai Peninsula, as well as natural gas imports from Israel, a close relationship with the United States and billions of dollars in American aid.

Emad Mekay and Adam Goldman contributed reporting.

The Chinese government on Friday approved a plan to raise the country’s statutory retirement age, currently among the lowest in the world, in a long-awaited but broadly unpopular effort to address the challenge of its rapidly aging population.

This is the first time China has raised its retirement age since the 1950s. It will be phased in gradually, starting on Jan. 1, 2025. The retirement age for men, previously 60, will increase in increments of several months before finally reaching 63 by 2040. The retirement age for women in white-collar jobs, previously 55, will rise to 58. Women in blue-collar jobs, who previously could retire at 50, will have to work until 55.

The decision would “maintain the momentum and vitality of economic and social development,” Wang Xiaoping, China’s Minister of Human Resources and Social Security, said at a news conference.

Policymakers and experts have for decades been calling for a change to the retirement age, noting that the previous rules dated from a time when life expectancy in China was much shorter, and fertility rates higher. They warned that maintaining the status quo would severely strain the country’s work force and pension funds, with large numbers of older Chinese retiring but fewer young people replacing them.

China’s working-age population has been falling since 2012, according to official statistics, with an average annual decrease of more than 3 million people. Last year, China had 297 million people over 60 years old, or about 21 percent of its population.

But the proposal faced broad opposition, from older workers as well as from younger ones, who worried that an expanded work force would mean even tougher competition for jobs. China also lacks a strong social safety net, and age discrimination by employers is common, leaving many blue-collar workers to worry that they would be left without work but unable to draw on pensions.

In a sign of how sensitive the issue was, the government has promised before to raise the retirement age, only to backpedal in the face of public outcry.

In some ways, now is a particularly bad time for the government to follow through. China’s economy is still struggling to recover from the coronavirus pandemic. Youth unemployment remains high. Local governments, which administer pensions and other government benefits, have already cut back on payouts in recent years — at times setting off protests.

But the government most likely realized it had no more time to delay, said Alfred Wu, a professor of public policy at the National University of Singapore, who noted that he was surprised by the speed with which the government had ultimately made its move. (The legislature had announced earlier this week that it was reviewing a draft law, and there was no public comment period as is common for many laws.)

“I think China has lost a lot of golden opportunities” to make the decision more palatable to the public, Professor Wu said. “Maybe another way is just to release it. Let people accept it, and they move forward.”

Acceptance did not seem to be the initial public reaction.

On Chinese social media, where several hashtags about the decision led the trending topics, users complained that they would have to wait even longer to receive benefits that they decried as too low anyway.

Others worried that the pension funds would be even more depleted by the time they finally got to retire. The government-backed Chinese Academy of Social Sciences has previously projected that China’s pension funds would run out of money by 2035.

The decision announced Friday will also lengthen the time people must pay into pension funds before becoming eligible to receive a monthly pension, to 20 years from 15.

In an apparent effort to ease people’s concerns, the plan included what it described as a flexible option for people to retire up to three years earlier — that is, at the current retirement age — if they had fulfilled the 20-year contribution requirement.

“The decision reflects respect for individual wishes. It does not force everyone to reach the new statutory retirement age,” Xinhua, the official news agency, quoted a professor at Zhejiang University, Jin Weigang, as saying.

The raising of the retirement age by three years for most workers was less than the five years many had expected.

The government also promised to improve the system for paid annual leave, fight age discrimination and offer earlier retirement for workers in physically demanding jobs.

And it said it would work to establish universal child care — another step that could be crucial for ensuring that the delayed retirement age does not inadvertently worsen the population aging problem, by discouraging people from having children. Currently, many retirees take primary responsibility for raising their grandchildren.

Besides those reassurances, the government also had another tool to ensure that public backlash did not spiral out of control: censorship.

On Chinese social media, major state media outlets controlled which comments on their posts about the news could be seen. “Got it,” said one of the top comments under a post by the People’s Daily, the Communist Party’s mouthpiece.

Siyi Zhao contributed research.

Over the last couple of weeks, Pope Francis hoisted himself up on sore legs dozens of times. As he crisscrossed large swaths of the Asia Pacific region, he shuffled from cars to his wheelchair, from the wheelchair to makeshift papal thrones and on and off many planes as hot, tropical winds blew on the tarmac.

The trip amounted to the longest and farthest reaching yet for Francis, and at 87, some of his supporters fear it may be one of his last. But that he flew thousands of miles to Asian countries with relatively small Catholic populations, braving oppressive temperatures and high levels of humidity and pollution, underlined Francis’s commitment to building a church with a less Eurocentric future.

“The long distance, the fatigue, the challenges,” said Cardinal Michael Czerny, a close aide to Francis. “They are part of the message.”

Francis’ papacy has from the start been one of symbols: the small modest cars he’s used, kissing the feet of criminals, the Casio watch he wears. His destinations are as much a part of his teachings as his homilies, and this trip to Indonesia, Papua New Guinea, East Timor and Singapore was seen as part of how he has defined his papacy.

The purpose, he has long made clear, is to emphasize outreach and inclusivity. As he visited remote, tropical villages in Papua New Guinea, he put in practice his pledge to embrace what he calls the church’s “peripheries,” faraway, minority or poorer Catholics.

Asia, home to two thirds of the world’s population and an increasingly central player on the global stage, has long been a focus for Francis. He has been unable to go to China, where the church sees enormous potential but also faces great obstacles. China and the Vatican do not have official diplomatic relations even though Francis signed a groundbreaking and heavily criticized deal with the Chinese government on the appointment of bishops.

But with this trip he completed a tour of much of its surrounding region, visiting eleven countries around China. Francis departs his last stop, Singapore, on Friday, and then returns to the Vatican.

For his supporters, the fact that he did all of this at 87, after suffering a series of health issues, and saying that traveling had become harder, only made his endeavor more meaningful. As Pope Francis shook hands, and gleefully waved to the crowd, bishops held on to their purple skullcaps in the wind and looked on at him in admiration.

“I don’t know where he finds the energy,” said Bishop Jozef Roszynski of Wewak, in Papua New Guinea. “Yesterday he was on the trolley going all over the place.”

Pope Francis showed moments of great vitality during the trip. He spoke off the cuff, cracking jokes about crocodiles, cats, dogs and the Devil. In Papua New Guinea he flew over vast timber forests on an Australian Air Force plane to a remote Pacific town.

There, he put on a feathered headdress, then traveled by the sandy coastline to an even more remote school run by Argentine missionaries, where he drank mate tea. In the capital of a country that has seen bloody tribal violence, he sat in a scorching stadium and asked the thousands of people gathered if they preferred harmony or confusion. Then, like an entertainer, he said, “I can’t hear you,” to get a louder, more emphatic response.

He also had times in which he appeared tired or less engaged. As he gave a speech to leaders in the capital, Port Moresby, and praised a land “so far from Rome but so close to the heart of the Catholic Church,” he stopped several times to cough.

In Jakarta, he did not lead the Mass at the stadium, which would involve lots of standing, and as he arrived in Port Moresby, he appeared to lose his balance for a moment.

But he went on, and locals appreciated the effort.

“Now I feel I am part of a universal church, and that we are not just somewhere remote,” said Justin Ain Soongie, the bishop of Wabag, a small town in the highlands of Papua New Guinea, as he exited a meeting with Francis at a local church with enormous fans slicing through the warm air.

Pope John Paul II visited Papua New Guinea in 1984, at age 63, but Francis, Bishop Ain Soongie pointed out, “took the risk to come especially at this age and on a wheelchair.” By traveling so far away, he added, “He is living what he is saying.”

In East Timor, about half of the country attended a Mass presided over by Francis, and people climbed on roofs to get a glimpse of him. Huge billboards with Francis’ face appeared among sheet metal shacks in impoverished suburbs and in the lush gardens of Singapore. Around the region, faithful wearing suits, dusty T-shirts or straw skirts waved Vatican flags. In Papua New Guinea, troubled by local rivalries, people came together to see the pope, some after walking across the forest for days.

Chris Anowan, 54, a school principal there, traveled to the city of Vanimo one week before Francis’ visit, because the dirt road connecting his village to the town had dried out from rains, and he did not know how long it would remain passable.

“For me, it’s like seeing Saint Peter,” said Mr. Anowan, as he waited for Pope Francis with rosaries and plastic bottles of water ready to be blessed. “It’s the first time a pope comes here,” he said, referring to Vanimo.

Some of those who traveled alongside the pope from Italy, a country where the number of practicing Catholics has steadily declined, were impressed by the local fervor.

While the number of Catholics in Asia is growing at a slower pace than that of Africa, the church is lively here. Countries like Indonesia, where for centuries Europeans went to evangelize, are now exporting missionaries.

“If the Catholic Church wants to have a place in the future it cannot be excluded from Asia,” said Andrea Riccardi, the founder of the Sant’Egidio Community, a Catholic group close to Francis.

Francis, who as a young priest in the Jesuit religious order had wanted to go as a missionary to Japan to follow the centuries-old pursuit of his predecessors, has acknowledged the region’s importance.

He has worked to improve the Catholic Church’s relations with Vietnam as well as China and named over a dozen Asian cardinals — including from countries like East Timor and Singapore, which had never had one.

But Asia, experts say, is also a testing ground for the church, as Catholics here often coexist with bigger religious groups.

“The Catholic Church and the papacy are learning to confront themselves with a world in which Christianity is a minority,” said Massimo Faggioli, a professor of theology at Villanova University, in Pennsylvania. Asia, he said, is “the most challenging continent.”

Indonesia, the main stop of the trip, encapsulated this reality. It is the country with the world’s largest Muslim population, but also home to millions of Christians. Coexistence is at the heart of its identity, even though episodes of intolerance against Christians persist.

There, Pope Francis signed an agreement with Nasaruddin Umar, the leader of Southeast Asia’s biggest mosque, as the two men exchanged affectionate embraces.

“With the pope coming here, other countries will see how we live peacefully together,” Catur Rini, 63, a Muslim woman who had come to the event in Jakarta’s Istiqlal Mosque with her Catholic high school friend. “He will share it with the world.”

In the last stop of his trip in Singapore, a financial powerhouse with only a small Catholic population, Francis filled the city-state’s national stadium with worshipers for a Mass.

Kat Calimag, 32, a veterinarian, had gone there just a few months ago to watch Taylor Swift perform. This time, too, fans screamed and cried.

“Same energy,” she said.

For a brief moment two years ago, Pakistan seemed finally on the verge of defeating polio. One of only two countries in the world where the virus remains endemic, Pakistan recorded no new infections for a little over a year starting in 2021 — the longest virus-free stretch the country had ever experienced.

But since then, polio has roared back, spreading beyond its traditional hot spots to areas once largely untouched by the virus.

Last week, health officials reported the first polio case in the capital, Islamabad, in 16 years. This month, environmental monitoring detected the polio virus in sewage samples from several major cities, including Peshawar and Karachi, Pakistan’s largest city, where millions live in crowded, unsanitary slums.

And the virus has spread to a new epicenter in Balochistan, an arid, restive province in the southwest hundreds of miles from the virus’s former focal point in northwestern Khyber-Pakhtunkhwa Province.

On Monday, Pakistan began a weeklong nationwide polio vaccination campaign involving 286,000 health workers — the largest public health surveillance network in the world — aimed at vaccinating 30 million children under 5. The campaign, taking place across 115 of the country’s more than 165 districts, is part of the government’s renewed billions-dollar effort to contain the spread of the virus.

“I am hopeful that polio will be eradicated in the coming years and months through coordinated efforts,” Shehbaz Sharif, the country’s prime minister, said on Monday. “Polio will be driven out from the borders of Pakistan, never to return.”

The resurgence of polio in Pakistan is part of a global comeback of the virus, a highly contagious and sometimes deadly illness that once paralyzed hundreds of thousands of children worldwide every year. After vaccines were introduced in 1955, the number of cases dropped around the globe by more than 99.9 percent.

But after health authorities decided in 2016 to pare down the oral polio vaccine, the virus roared back. Since then, cases of vaccine-derived Type 2 polio have increased tenfold. This month alone, at least eight countries were battling polio outbreaks.

In Pakistan, the health authorities face an array of challenges.

Not only is the country home to difficult terrain, nomadic populations and poor infrastructure where polio thrives, but misinformation is also rampant, which has led to widespread distrust of vaccines.

Conservative religious scholars and militant groups have falsely asserted that the vaccination campaign is a Western conspiracy to sterilize Muslims, or that the vaccines contain ingredients derived from pigs, which are forbidden in Islam. Such claims have prompted entire communities to refuse vaccination.

Another problem: militants who attack vaccinators. This year, 15 people, mostly police officers, have been killed and 37 injured during vaccination campaigns, according to officials.

“Police officers are always easy targets, but those protecting polio vaccination teams are even more vulnerable,” said Muhammad Jamil, a Peshawar police officer.

Authorities in neighboring Afghanistan (the only other country where polio is endemic) have reported 18 polio cases so far this year, according to the Global Polio Eradication Initiative. These cases are primarily concentrated in southern provinces like Kandahar and Helmand, regions that border Pakistan and where cross-border movement of population makes the risk of transmission particularly high.

Global health officials planned to coordinate the vaccination drive across both countries in September and October to ensure those in nomadic communities are not missed. But logistical issues delayed Afghanistan’s campaign.

Pakistani health workers have also reported facing pressure from parents and local leaders to falsely mark children’s fingers with indelible ink, indicating they’ve been vaccinated even if they haven’t — a practice that health officials say has significantly contributed to the virus resurgence. Many vaccinators also do not report when families refuse to be vaccinated, fearing backlash if authorities take action against resistant families or tribes, health officials said.

In Pakistan’s tribal areas, polio vaccination has been used as a bargaining chip for local leaders desperate for better government services in a region that the authorities have historically overlooked.

On Monday, some communities in Khyber Pakhtunkhwa Province announced that they refused to vaccinate their children unless their demands — including improved infrastructure and support for people returning to villages after they were displaced during military operations — were met.

Protester leaders ordered community members to comply with the boycott, threatening to fine any family who violates it.

“We may not know whether the vaccine harms our children or not,” said Malik Shamshad, a leader of one of the boycott campaigns. “But we know that the government comes under pressure and resolves our problems when we refuse to vaccinate the children.”

Still, many health workers remain undeterred. Early Monday in Peshawar, a bustling city near Pakistan’s border with Afghanistan, around three dozen health workers gathered at a government health facility to start the week’s vaccination drive. With colorful scarves over their heads and shoulders, the workers recited verses of the Quran while police officers stood nearby.

“We know it’s a long battle,” said Firdos, 33, a health worker who preferred to go only by her surname for fear of retaliation. “But for the sake of our children, we have to keep going.”

North Korea on Friday revealed a weapons-grade uranium-manufacturing site for the first time, as its leader, Kim Jong-un, flaunted his expanding nuclear weapons program ahead of the United States presidential election in November.

Mr. Kim recently visited a centrifuge plant — until now, a highly guarded component of the country’s nuclear weapons program — and urged his engineers to expand their production of highly enriched uranium to build “exponentially” more nuclear weapons, the North’s official Korean Central News Agency reported on Friday.

The news agency carried photos of Mr. Kim inspecting what looked like a modern plant packed with long rows of centrifuges, devices used to enrich uranium. The images will be pored over by foreign governments for intelligence at a time when tensions are rising with South Korea and when the Biden administration has pivoted the United States’ nuclear strategy to focus on possible coordinated threats from China, Russia and North Korea.

Although North Korea in 2010 showed a centrifuge plant to a visiting team of former U.S. officials and academics, including a former director of Los Alamos National Laboratory, Siegfried Hecker, the report and photos published in state media on Friday were the first time that the country has unveiled such a facility to the wider world.

A series of resolutions adopted by the United Nations Security Council ban North Korea from developing nuclear weapons. But the country has ignored them by producing nuclear bomb fuel and conducting underground tests of six nuclear devices and tests of ballistic missiles capable of delivering nuclear warheads.

After his 2010 visit to Yongbyon, Dr. Hecker said that the uranium-enrichment facility there appeared to contain about 2,000 gas centrifuges. But North Korea is widely believed to operate other centrifuge plants in other, secret locations. Experts suspect that Kangson, just outside Pyongyang, is one such location. In June, Rafael Grossi, the director-general of the International Atomic Energy Agency, said that the Kangson complex shared characteristics with the centrifuge enrichment facility at Yongbyon.

North Korea on Friday did not reveal the location of the plant Mr. Kim visited or its production capacity.

But Mr. Kim “stressed the need to further augment the number of centrifuges in order to exponentially increase the nuclear weapons for self-defense,” state media reported on Friday. Mr. Kim also said his country was introducing “a new-type centrifuge” that would help expand its weapons-grade nuclear materials.

The so-called North Korean nuclear crisis began in the early 1990s when the United States began accusing the country of producing plutonium from the spent fuel from a small Soviet-designed nuclear reactor in Yongbyon. It was later revealed that the country was securing another type of fuel for atomic bombs: highly enriched uranium.

North Korea conducted six nuclear tests between 2006 and 2017. In June, the Stockholm International Peace Research Institute estimated that North Korea had built roughly 50 nuclear warheads and had enough fissile material to build about 40 more.

The country’s nuclear force is especially dependent on missiles as delivery vehicles because it lacks advanced warplanes or submarines from which to launch them. Under Mr. Kim, North Korea has diversified its arsenal of missiles, testing various short- and long-range ones, as well as expanding its production of bomb fuel. North Korea launched several short-range ballistic missiles off its east coast on Thursday in its first missile tests in more than two months.

In recent months, Mr. Kim has emphasized the production of additional short-range ballistic missiles. North Korea has supplied Russia with some of these missiles for use in the war in Ukraine, researchers have said.

North Korea has also asserted that many of its short-range ballistic missiles were capable of carrying nuclear warheads. In August, it said it was deploying 250 launchers of such nuclear missiles near the border with South Korea. On Friday, North Korean state media quoted Mr. Kim as saying that his country needed more nuclear materials to make more such short-range “tactical nuclear weapons.”

Mr. Kim’s visit to the uranium-enrichment factory came as the campaign for the November presidential election in the United States heats up, in which North Korea has become one of the foreign policy issues under discussion.

“It’s nice to get along when somebody has a lot of nuclear weapons,” the former president Donald J. Trump said of Mr. Kim when he accepted the Republican Party’s presidential nomination in July. “I think he misses me, if you want to know the truth.”

During a presidential debate on Tuesday, his Democratic rival, Vice President Kamala Harris, criticized Mr. Trump for his personal rapport with Kim.

“It is well known he exchanged love letters with Kim Jong-un,” she said. Addressing Mr. Trump, she added, “And it is absolutely well known that these dictators and autocrats are rooting for you to be president again because they’re so clear they can manipulate you with flattery and favors.”

On Friday, the South Korean government said it strongly condemned North Korea’s nuclear weapons program, including its short-range tactical nuclear weapons. It said it would deal with the North Korean nuclear threat by strengthening the alliance with the United States. In the past couple of years, the allies have begin preparing joint plans to counter a possible nuclear attack from the North.

President Biden appears on the verge of clearing the way for Ukraine to launch long-range Western weapons deep inside Russian territory, as long as it doesn’t use arms provided by the United States, European officials say.

The issue, which has long been debated in the administration, is coming to a head on Friday with the first official visit to the White House by Britain’s new prime minister, Keir Starmer.

Britain has already signaled to the United States that it is eager to let Ukraine use its “Storm Shadow” long-range missiles to strike at Russian military targets far from the Ukrainian border. But it wants explicit permission from Mr. Biden in order to demonstrate a coordinated strategy with the United States and France, which makes a similar missile. American officials say Mr. Biden has not made a decision, but will hear from Mr. Starmer on Friday.

If the president approves, the move could help Ukraine hold the line after it seizes Russian territory, as it did during its surprise incursion into Russia’s Kursk region. But Mr. Biden has hesitated to allow Ukraine to use American weapons in the same way, particularly after warnings from American intelligence agencies that Russia could respond by aiding Iran in targeting American forces in the Middle East.

On Thursday, White House officials insisted there was no imminent decision on the use of the American-made surface-to-surface Army Tactical Missile Systems — known as ATACMS. But Mr. Biden himself has signaled that a loosening of restrictions is coming. He was asked on Tuesday whether he was ready to grant the increasingly insistent requests from President Volodymyr Zelensky of Ukraine.

“We are working that out right now,” he said.

If Mr. Biden permits the British and French to go ahead, and if he follows in coming weeks by allowing the use of the ATACMS, it could well be his final acceleration of the military aid to Ukraine.

Quietly, Republican leaders in the Senate, especially Mitch McConnell, the minority leader, have been urging an aggressive response — a sharp split with former President Donald J. Trump, who refused in Tuesday night’s presidential debate to declare that he wants Ukraine to win, or to say that Russia should get out of the 20 percent or so of Ukraine it has taken since war began.

On Thursday, President Vladimir V. Putin of Russia issued an unusually specific warning to the West, noting that the Ukrainians alone cannot operate the long-range missiles, because they require Western technical help and satellite guidance.

“This will mean that NATO countries — the United States and European countries — are at war with Russia,” Mr. Putin said, according to a report by the Kremlin. “And if this is the case, then, bearing in mind the change in the essence of the conflict, we will make appropriate decisions in response to the threats that will be posed to us.”

For the United States, assessing how much to believe Mr. Putin has been a difficult task. Over nearly 31 months of war, the pattern has been clear: At every stage, Mr. Biden has been concerned that providing new weapons to Ukraine, or allowing Ukraine’s military to shoot into Russian territory, would cross one of Mr. Putin’s red lines.

In the opening months of the war, Mr. Biden was reluctant to provide HIMARS artillery to Ukraine, then M1 Abrams tanks, F-16 fighter jets and short- and then longer-range ATACMS. But in each case, as the administration discovered Mr. Putin appeared less eager to escalate the war than initially believed, it loosened the reins.

In the spring, for the first time, Mr. Biden approved allowing Ukraine to fire at Russian artillery and other targets just over the Russian border, to avoid giving Mr. Putin’s forces a haven for attacking cities and towns around Kharkiv. That permission was later expanded. But striking the border areas is essentially a defensive operation. Senior White House officials say there remains worry about using American ATACMS to strike more than 60 or so miles into Russia.

In classified briefings, American intelligence officials have expressed deeper concerns about direct, visible American participation in Ukraine’s move to seize and hold positions near Kursk. There are indications, they have warned, that Russia could provide technological help that would allow Iran and its proxy forces to attack American forces in the Middle East. The administration this week accused Iran of shipping missiles for the first time to Russia for use in the war, an accusation the government in Tehran has denied.

In a series of meetings with senior administration officials in recent weeks, Ukrainian officials have been arguing that their seizure of actual territory inside Russia demonstrates that U.S. fears of crossing Russian red lines were overblown. The United States, those Ukrainian officials argue, should allow Kyiv to use American weapons to strike deeper into Russia.

Emerging from one of those meetings in Kyiv on Wednesday, Secretary of State Antony J. Blinken told reporters that he and Mr. Zelensky, joined by the British foreign minister, David Lammy, “discussed long-range fires, but a number of other things as well.”

“We’re going to take what we learned back to President Biden in my case, and the prime minister in David’s case. The two of them will meet in just a few days’ time in Washington to discuss how our countries will continue to support Ukraine.”

To a growing number of military analysts and former U.S. officials, the administration’s reticence makes no sense, especially since, they say, Ukraine’s incursion into Kursk has yet to elicit an escalatory response from Moscow.

“Easing the restrictions on Western weapons will not cause Moscow to escalate,” 17 former ambassadors and generals wrote in a letter to the administration this week. “We know this because Ukraine is already striking territory Russia considers its own — including Crimea and Kursk — with these weapons and Moscow’s response remains unchanged.”

Senior Ukrainian officials were at the Pentagon two weeks ago making a similar argument to Defense Secretary Lloyd J. Austin III. Ukraine’s new defense minister, Rustem Umerov, argued that the Kursk incursion showed that Russia’s red lines were simply bluffs that had slowed the Western effort help Ukraine.

With the Kursk incursion, Mr. Umerov argued, Ukraine has demonstrated it can invade, and even occupy, Russian territory without igniting World War III, according to two officials.

But American officials say it is too early to reach that conclusion, because there are many ways for Mr. Putin to retaliate. During the meeting, Mr. Austin asked Mr. Umerov several questions about which sites inside Russia Ukraine would target, probing to make sure Ukraine would focus on military sites like airfields, but not power plants or other civilian infrastructure. Mr. Austin also queried his Ukrainian counterpart about what objective such targeting could accomplish.

Mr. Austin continues to believe that the use of U.S. weapons for long-range strikes into Russia won’t turn the tide of the war, in part because there are not enough ATACMS — or British and French missiles — to sustain an attack.

At Ramstein, a U.S. air base in Germany, last Friday, Mr. Austin added that loosening the reins on Ukraine’s use of ATACMS would not resolve one of the biggest problems facing Ukrainian cities and troops — so-called glide bombs launched from Russian attack planes deep inside Russia.

“As we look at the battlefield currently, we know that the Russians have actually moved their aircraft that are using the glide bombers beyond the range of ATACMS,” Mr. Austin said.

The United States has already supplied Ukraine with several hundred of the long range ATACMS, but its stockpiles are running low. American officials are concerned they could not supply enough of those munitions to seriously damage a wide range of Russian targets.

Mr. Umerov’s counter — made during the Pentagon meeting, officials said — was that even if the ATACMS are not a game changer, they can still be used to good effect to hit Russian sites inside Russia and to disrupt Russian logistics.

“The strikes would help degrade Russian military capabilities, and Russia already uses Iranian, Chinese and North Korean weapons and components against targets in Ukraine,” said Seth G. Jones, a senior vice president with the Center for Strategic and International Studies.

Meanwhile, Maj. Gen. Patrick S. Ryder, the Pentagon press secretary, told reporters on Tuesday that the new shipment of short-range ballistic missiles from Iran to Russia set a dangerous precedent that is likely to lead to more shipments.

“One has to assume that if Iran is providing Russia with these types of missiles that it’s very likely it would not be a one-time good deal,” he said. “That this would be a source of capability that Russia would seek to tap in the future.”

Reporting was contributed by Lara Jakes in Jerusalem, Edward Wong in Kyiv and Peter Baker in Washington.

Three Red Cross workers were killed and two wounded on Thursday when artillery fire struck a frontline aid distribution site in Ukraine on Thursday, the organization said.

The International Committee of the Red Cross workers were preparing to distribute wood and coal briquettes in the village of Viroliubivka, in the Donetsk region, when they were hit, the group said in a statement.

The aid distribution had not yet started and no residents were harmed, the Red Cross said. The supplies were intended to prepare residents for the cold winter nights that are soon to come.

The Red Cross said that its teams worked in the region regularly and that its vehicles were clearly marked.

Images from after the attack show a white truck with a large red cross centered on the side engulfed in flames, its cab on fire and plumes of black smoke billowing upward in an arch.

The wounded staff members were taken to receive medical attention, and one was in serious condition, the Prosecutor General’s Office of Ukraine said in a statement.

The attack took place amid increasing Russian bombardment around the eastern Ukrainian city of Pokrovsk, roughly 35 miles away, as Russian troops pressed ahead with an offensive aimed at capturing the strategic city. Conditions in the city have deteriorated, and residents who remain are largely without water or electricity.

The Red Cross denounced the shelling without assigning specific blame.

“I condemn attacks on Red Cross personnel in the strongest terms,” the president of the International Committee of the Red Cross, Mirjana Spoljaric, said in the statement. “It’s unconscionable that shelling would hit an aid distribution site. Our hearts are broken today as we mourn the loss of our colleagues and care for the injured.”

President Volodymyr Zelensky of Ukraine blamed Moscow for the deaths and called the attacks “another Russian war crime” in a statement paired with a photo of the Red Cross vehicle ablaze.

Last year was the deadliest on record for humanitarian aid workers, according to the United Nations, which reported 280 aid workers killed in 33 countries in 2023, a 137 percent increase over the year before.

For 2023 and the first half of 2024, the I.C.R.C. reported six staff members killed and 14 injured. That figure did not include Thursday’s casualties.

Pat Griffiths, a Red Cross spokesman in Ukraine, was at a conference in Kyiv when he received news of his colleagues’ deaths. Mr. Griffiths said he had recently traveled to the eastern Ukraine with the team whose members were killed.

“The feeling right now is grief,” he said. “And then because of our neutrality, all we can do is repeat the call, this urgent call, for all countries around the world, all parties to a conflict, to respect international humanitarian law, which is crystal clear that humanitarian workers and those who help, and ambulance drivers or first aid responders, are not targets.”

Asked if the group would cease operations in that area of Ukraine, another spokesman, Jason Straziuso, said the Red Cross monitors the security situation carefully and planned to do an analysis of the attack.

“We know that our work helping people close to the front lines of conflict is inherently dangerous,” Mr. Straziuso said. “We know we cannot avoid all risks. When people think about the risk humanitarian workers face, we hope they also think about the risk residents who live close to the front lines of conflict face.”

Maria Varenikova contributed reporting.