There is no sign on the door of the new Hamas political office in Baghdad, and the address is closely guarded. The same goes for the new Houthi office, a short drive away.

Iraqi government officials quietly allowed both Iranian-backed armed groups to establish a more permanent presence in Baghdad early this summer, after years of their representatives visiting. The shift, which Iraqi officials deny publicly even as photos of the groups in Iraq circulate on social media, comes as Iran has appeared to encourage its proxies from different countries to share military skills and even coordinate on targets.

The new offices reflect Iraq’s growing role in the shadow war between Iran, Israel and the United States.

For more than 20 years, since the U.S. invasion to oust the Iraqi dictator, Saddam Hussein, Iraq has struggled to maintain an uneasy balance between Iran, with which it shares a 1,000-mile border, and the United States, which still maintains about 2,500 troops in the country.

The balance has gradually shifted in favor of Iran. Iraq’s neighbor has worked steadily to amplify its geopolitical sway by expanding recruitment and funding of sympathetic forces inside Iraq. It is part of a larger effort by Tehran to build a regional bloc of Shiite power that extends to Lebanon with Hezbollah and to Yemen with the Houthis.

Iran in recent years pushed the Iraqi government to legitimize the country’s Shiite militias, some with loyalties to Tehran, as well as affiliated Sunni, Christian and Yazidi armed groups, by making them part of Iraq’s security apparatus. The Shiite forces have also created successful political parties, a coalition of which won enough seats in the 2021 election to choose the prime minister.

Against that backdrop of Iran’s rising influence, Iraq’s leadership acquiesced when the Houthis and Hamas wanted to open offices. Some Iraqi government officials, according to two of the people who spoke to The New York Times, say privately they are not thrilled about their new guests but did not have the power to block them, given the sway of the Iraqi political parties with ties to Iran.

The offices, mainly focused on developing links in Iraq, were established in June, according to the Iraqi and Western officials, as well as a member of an Iraqi armed group. They spoke on the condition of anonymity to discuss sensitive topics.

Iraq and Iran’s shifting relationship

The two new offices, one for Hamas, a Sunni group, and the other for the Houthis, a Shiite one, reflect how much Iraq’s politics have changed since the time of Mr. Hussein.

Although he was a Sunni Muslim, his regime suppressed Sunni Islamist movements like the Muslim Brotherhood with which Hamas was affiliated. He saw them as a potential threat to the hegemony of his Ba’ath Party. Many Iraqis remained leery of such groups long after Mr. Hussein’s fall, not least of all because of the rise of Sunni militants, including Al Qaeda in Iraq and, later, the Islamic State.

As for Shiite Muslim movements, they were violently suppressed under Mr. Hussein, who feared they would conspire to depose him. Members of such groups were either forced to flee or were imprisoned and executed.

Politics in Iraq today are dominated by Shiite parties, with a strong affinity to Iran. The shift has allowed foreign groups with ties to Iran to make inroads, bolstering what is known as Iran’s Axis of Resistance, its armed network across the Middle East that is dedicated to countering American and Israeli influence in the region.

Thomas Juneau, a professor of international relations at the University of Ottawa, said he and other academics had noticed a trend by Iran of encouraging armed groups from different countries to work together. They point to efforts by Iraqi and Lebanese groups on behalf of the government of Bashar al-Assad during the Syrian civil war.

There is “a growing institutionalization of relations between Iran’s partners in the Axis of Resistance,” he said. With that end in mind, he said, Tehran has created joint operation rooms and held regular meetings that bring their leaders together, efforts that have “intensified” since Hamas and its allies attacked Israel and the war in Gaza began.

One worry is that the presence of so many Iran-backed groups in Iraq could prompt Israel’s military to strike inside Iraq, further destabilizing the region.

Similarly, there are concerns that the Iraqi armed factions will team up more frequently with the Iranian backed proxies and time their attacks against Israel together. The Iraqi groups, along with the Houthis, claimed to jointly attack Israel eight times in June and three times in July, according to a strike calendar maintained by the Washington Institute, a D.C.-based think tank.

Cultivating local ties

Hamas opened its Baghdad office in Arosat, a middle-class neighborhood with a mix of two-story homes built in the 1970s and more recent construction, traversed by streets featuring a smattering of pizza cafes and furniture stores as well as new buildings, some still under construction. Parts of the area are controlled by Kata’ib Hezbollah, the most prominent and secretive of the Iraqi Shiite armed groups loyal to Iran.

The Hamas representative in Baghdad is Mohammed al-Hafi, a member of the Hamas bureau for Arab and Islamic Relations. Reached by phone last month in Baghdad, Mr. al-Hafi declined a request to speak with The Times, saying, “I do not have permission to speak with the media.”

Mr. al-Hafi, whose security detail in Iraq is provided by Kata’ib Hezbollah, has met with a number of Iraqi groups and individuals, both Shiite organizations associated with the Axis of Resistance and Sunni groups that share Hamas’s Muslim Brotherhood philosophy.

Hossam al Rubaie, a spokesman for Khadamat, an Iraqi Shiite political party affiliated with an armed group close to Iran, said he met with Mr. al-Hafi on several occasions. He said the office provides a way for Iraqis to have a direct link with Hamas.

Mr. al-Hafi is “a political figure, not a military figure, and having an office allows him to convey messages to Iraqi politicians directly, not through an intermediary,” said Mr. al Rubaie.

The opening of the Hamas office has also been a boon for the Iraqi Islamic Party, a Sunni party that shares its Muslim Brotherhood philosophy but has had little public presence in recent years. The Brotherhood, an international Sunni organization, has been condemned as terrorist by some Arab countries, but embraced by others.

Rashid al-Azzawi, the Iraqi Islamic Party’s chairman, said that the war in Gaza, which has been devastating for civilians, has brought sympathy from Iraqis of all religions and made them more open to Hamas’s having a presence. The group, he said, is seen as fighting for “a humanitarian cause.”

The reception to the Houthis has been especially warm among Shiites.

Since early July, the Houthi representative in Iraq, Abu Idris al-Sharafi, has met with Baghdad heavy hitters, including Qais al-Khazali, the leader of the Iranian-linked armed group Asa’ib Ahl al-Haq and the founder of its influential political wing. The Houthi representative also visited tribal leaders in rural southern Iraq, posting a video on the Houthis’ Iraq channel of a speech there in which he was wearing a ceremonial dagger and gesticulating vigorously as he urged them to wage jihad against Israel.

“The presence of a representative of the Houthis in Iraq is welcomed by all political parties in Iraq,” said Saad Al-Saadi, a member of the political leadership of Sadiqoon, the leading Shiite political party in Parliament. “Especially because they represent the Yemeni government and also represent an important part of the Axis of Resistance.”

Falih Hassan contributed reporting.

Ukraine has asked to use Western long-range weapons to strike deeper into Russia for months. It argues that it needs those weapons to hit military sites that house Russian warplanes and that launch missiles into Ukrainian cities.

Those entreaties were a major topic of discussion on Friday as President Biden met with Prime Minister Keir Starmer of Britain, who is trying to nudge the United States to give more latitude to the Ukrainians. Mr. Starmer would especially like Mr. Biden’s support for Britain to allow Ukraine to use British Storm Shadow missiles to fire farther into Russia.

Neither leader announced any policy changes after that meeting. Leaving the White House, Mr. Starmer told reporters, “We had a wide-ranging discussion about strategy,” and a White House summary of the meeting said that the two countries had “reaffirmed their unwavering support for Ukraine.”

Mr. Biden has been reluctant to approve deep strikes in the past, fearing escalation with Russia. But in May, he allowed Ukraine to fire a number of U.S.-supplied weapons just over the border to attack Russian military bases from which attacks into Ukraine have been launched, and he later expanded that permission.

Now, Ukraine wants long-range weapons. It is also seeking permission to use weapons it already has to hit targets deeper in Russia. President Vladimir V. Putin of Russia warned on Thursday that if the United States and its allies allowed that, they would put his country “at war” with NATO.

The debate in the U.S. administration centers, in large part, on these weapons.

Storm Shadows and SCALPs

Britain and France have already sent Ukraine air-launched cruise missiles that, so far, have struck Russian targets in Crimea and in the Black Sea. These missiles have a range of about 155 miles and have been fired from Ukraine’s aging fleet of Soviet-era and Russian-designed fighter jets.

They are known as Storm Shadows in Britain and SCALPs in France (and are virtually the same model).

Britain is eager to allow Ukraine to use the Storm Shadows to strike farther into Russia. Mr. Starmer was hoping to receive Mr. Biden’s approval of that plan so the allies could present a united front.

France has previously expressed support for Ukraine’s deep strikes into Russia, but only on military targets directly linked to Moscow’s war efforts in Ukraine.

Some analysts expect the United States to follow a pattern it established with Ukraine’s previous requests for weapons, like Abrams tanks, F-16 fighter jets and Patriot air defense systems: After long deliberations, Washington eventually allows its allies to move first in providing Ukraine with new capabilities or permissions, and then sometimes follows suit.

ATACMS

The Army Tactical Missile Systems, known as ATACMS (pronounced “attack ’ems”), are American-made long-range missiles that are filled with 375 pounds of explosives and, depending on the model, can strike targets up to 190 miles away. The United States supplied Ukraine with ATACMS last year, but the Biden administration has so far withheld its approval for their use across the border into Russia.

Russia has now moved 90 percent of its air bases that house bomber jets out of ATACMS range, U.S. and European military officials said, in anticipation that Ukraine could soon be allowed to fire the missiles across the border.

Originally developed in the 1980s to destroy Soviet targets far behind enemy lines, ATACMS could also strike Russian ground-based air-defense systems that target Ukraine’s newly furnished fleet of F-16s, experts said.

JASSMs

Joint Air-to-Surface Standoff Missiles, or JASSMs (pronounced “jazz ’ems”), are air-launched cruise missiles with a range of about 230 miles. These have not yet been provided to Ukraine, but a U.S. official said that the Biden administration was considering sending them.

The weapons carry 1,000-pound warheads and can be fired from F-16s. This means that with JASSMs, Ukraine could strike military targets well within Russian territory without leaving Ukrainian airspace.

Ukraine is believed to have received around a dozen American-made F-16s this summer, though officials have not said exactly how many.

The U.S. official said that even if Mr. Biden approved sending JASSMs to Ukraine, delivery might take months, and it is unclear whether Mr. Biden would allow Ukraine to fire these missiles into Russia.

Eric Schmitt and Aurelien Breeden contributed reporting.

The driver braked hard and came to a screeching stop. Fans gasped, awed at the precision. Referees conferred, making sure all was aboveboard.

“We’re ready to rumble,” said Markus Chencinsky, 50, a driver from Vienna.

Despite the adrenaline, the speed and the thousands of eyes, this was not a demolition derby or a stunt-driving expo.

Instead, the competitors facing off on Saturday in a central square of Frankfurt were the captains of Europe’s tram systems. They had come to Germany’s financial hub to vie for the trophy at the 11th European Tramdriver Championship.

The annual public transit jamboree might best be described as tram dressage. The drivers coaxed their commuter chariots through an obstacle course meant to test their whimsy, mettle and precision.

“We try to mirror the entire range of skills a driver should have,” said Wieland Stumpf, the president of the championship.

Some events focused on safety: Drivers had to emergency brake at a precise spot, just as if a cyclist had swerved in front of them. Another tested their ability to multitask: Could they remember a series of symbols that appeared on mock traffic signs?

A few challenges evaluated the smoothness of their touch: Drivers had to come to a stop so gently that water did not slosh out of a bowl that was filled to the brim. (A front-mounted camera showed every lost droplet: the less spilled, the more points.)

One test was downright counterintuitive: Tram billiards, in which a driver steers the vehicle to gently knock a pool cue attached to a stand into a billiard ball on a table. (The highest possible score for the billiards portion was 500 points, awarded if the ball rolled to a stop right in the middle of the table.)

“It’s not often you’re trying to hit something with your tram,” joked Victoria Young, 39, of Edinburgh. “You’ve just got this feeling inside you that says, ‘I should be stopping now.’”

Tram driving is surprisingly gentle — even for all its mandated collisions. It stands out from other niche sports involving heavy machinery.

Take lawn mower racing, which is half motocross, half go-karting. Or snowmobile watercross, in which competitors drive their vehicles onto water, almost as if they were Jet Skis.

Tram driving’s closest cousin may be the backhoe rodeo, in which operators perform delicate tasks like placing helmets onto cones using their machines’ hulking yellow claws.

But while backhoe contests typically attract construction workers, the tram competitions draw public transit fanatics with strong opinions.

“Buses are the worst,” said Nosa Tasslimi, 27.

How about the metro? “I like subways more than trams, but there are no subway competitions.”

Mr. Tasslimi even designed a soccer fan-style scarf in support of the team from Oradea, a Romanian city to which he had no obvious connection. He and his friends — who were also not Romanian — wore the scarves with pride.

Earlier this year, transit systems held internal tournaments to select their 2024 teams. With the exception of Kyiv, Ukraine, every competing city had a male and female representative, said Mr. Stumpf, the championship’s president. (Most Ukrainian men have been barred from traveling internationally during the country’s war with Russia.)

Many of the 26 competing teams thought that Frankfurt, as the home team, would have an advantage. But Mr. Stumpf thought those concerns were unfounded, adding that trams are mostly alike in terms of design and control boards. Only one host city — Brussels, in 2019 — has ever won first place.

The teams spent part of Friday racing back and forth on unused tracks near a depot, getting a feel for their vehicles. For some drivers, it was a little bit like the first few minutes spent adjusting the mirrors and the seat in a rental car.

“We have been practicing,” said Virendra Mohan, 47, of Rotterdam, the Netherlands. “But the other teams have also been practicing, too.”

Mr. Mohan had been testing his precision stops. But the drivers representing Edinburgh, which was competing for the first time, had trained two hours a week for about six weeks. Dougie White, the team’s coach, rigged up a pool table in the depot. In Frankfurt, they had to get a feel for different pedals, one of which was where the horn would normally be on their trams.

“That’ll be the biggest challenge for them,” Mr. White said: “Driving a different tram.”

Vienna, next year’s host, is already brainstorming ideas for the competition. The pressure is on: It could host teams from Asia, the Americas and beyond for the first truly world championship. So far, it’s keeping the details of the next championship largely under wraps.

Christian Ludwig, an event manager with Vienna’s public transit operator, said he came to scope out ideas. “We are a tram bowling city,” Mr. Ludwig hinted, alluding to a beloved event that was not included in this year’s rotation. “That I can say already.”

The tournaments are partly meant to laud the almighty tram, which has the feel of a bus but travels along set tracks with the precision of a subway. They have been embraced by many cities looking to limit cars.

Mostly, though, the competition is a celebration of the drivers. It’s also an attempt to market the profession.

Cities across Europe are struggling to recruit drivers, Mr. Stumpf said, and the job can be thankless. Commuters rarely acknowledge the drivers’ efforts to ensure the trams are safe, reliable and jostle-free.

Still, many drivers take immense pride in their work. At the competition, some wore socks bearing their transit company’s logo. A few wore large tram pins. Elinor Svensson, 23, painted her nails yellow and blue, the colors of the Swedish flag.

“I want my passengers to be impressed,” she said.

As she loops around Stockholm, she said, she tries to drive so smoothly that her riders take notice. But as a passenger, “you don’t really notice it when it’s good,” she admitted. “You notice it when it’s bad.”

Ms. Svensson often imagines her riders at work or in classrooms, or perhaps talking to their doctors or friends.

“If I wasn’t here,” she said, “they wouldn’t get to where they needed to be.”

That’s part of what made the weekend so joyful. For two days, the drivers were celebrities — of public transit, that is.

In the end, Budapest came out on top. It was faster, smoother — maybe luckier.

“We were training hard,” said Ákos Bodnár, 25, a member of Budapest’s team who also won the runner-up award for best driver. “We can’t believe it.”

When the news broke that Ye, the rapper formerly known as Kanye West, would be performing in China on Sunday, the elation of many of his fans was mixed with another emotion: confusion.

Why would the notoriously prickly Chinese government let in the notoriously provocative Ye? Why was the listening party, as Ye calls his shows, taking place not in Beijing or Shanghai, but in Hainan, an obscure island province?

Under a trending hashtag on the social media site Weibo on the subject, one popular comment read simply “How?” alongside an exploding-head emoji.

The answer may lie in China’s struggling economy. Since China reopened its borders after three years of coronavirus lockdowns, the government has been trying to stimulate consumer spending and promote tourism.

“Vigorously introducing new types of performances desired by young people, and concerts from international singers with super internet traffic, is the outline for future high-quality development,” the government of Haikou, the city hosting the listening party, posted on its website on Thursday.

But it is unclear whether the appearance by Ye — who would be perhaps the highest-profile Western artist to perform in mainland China since the pandemic — is part of a broader loosening or an exception.

Even before the pandemic, the number of big-name foreign entertainers visiting China had been falling as the authorities tightened controls on speech. Acts such as Bon Jovi and Maroon 5 had shows abruptly canceled, leading to speculation that band members’ expressions of support for causes like Tibetan independence were to blame. Justin Bieber was barred from China in 2017 over what the Beijing city government, without specifying, called “bad behavior.”

Ye might have seemed like a no-go too. The Chinese authorities declared war on hip-hop in 2018, with the state news media saying that artists who insulted women and promoted drug use “don’t deserve a stage.”

But in Ye’s case, objections to hip-hop may have been outweighed by the potential payoff — especially for Hainan.

For years, the Chinese government has sought to turn Hainan, an island roughly the size of Maryland or Belgium, into an international commercial hub. It offers visa-free entry and duty-free shopping, and has pledged to attract more world-class cultural events.

Sheng Zou, a media scholar at Hong Kong Baptist University, said enforcement of censorship was capricious. “When it comes to Ye, I guess his celebrity status may outweigh his identity as a hip-hop artist.”

For Ricardo Shi, 25, an employee of a tech company in Shenzhen, the chance to see Ye was worth spending $700 on plane tickets for a two-day trip to Haikou. “It’s been so long since he last came to China,” he said. (Ye performed in Beijing and Shanghai in 2008.) “It’s a rare opportunity to be there in person.”

Ye, who is touring to promote “Vultures,” his new album series with the singer Ty Dolla Sign, has praised China. He told Forbes in 2020 that the country “changed my life.” He lived in the city of Nanjing as a fifth grader, when his mother was teaching English there.

And issues that have led Western brands to cut off collaborations with Ye and alienated many American fans, like his antisemitic and homophobic comments, are of less concern to Chinese officials.

Still, no artist can escape political scrutiny altogether.

A photo circulating on Chinese social media showed officials gathered around a conference table, before a screen that read: “Haikou Municipal Bureau of Tourism, Culture, Radio, Television and Sports ‘Kanye West World Tour Audiovisual Concert’ Risk Assessment Meeting.”

Reached by telephone, an employee at the bureau could not confirm the photo’s authenticity, but said that similar meetings were routine before large-scale events.

“These things, in my opinion, are a kind of test,” the employee, who gave her surname, Wang, said of the Ye event. “In the future there will be more foreigners coming to Hainan for similar concerts. As long as they provide positive energy, we’ll do it.”

No one has announced what songs Ye will play. Set lists must be preapproved by censors.

A week before officials announced Ye’s Hainan stop, a listening party planned for Taiwan was abruptly canceled. The Taiwan organizers cited “unforeseen circumstances.” It is unclear whether the cancellation was related to Ye’s show in China, which claims sovereignty over Taiwan.

A publicist who has worked with Ye on past listening parties did not respond to a request for comment.

Even after Ye passed the Chinese censors, some complained that he should not have. A string of submissions to Haikou’s public complaints website objected to his lyrics and personal behavior, with one user declaring them “inconsistent with our country’s cultural and social values.”

Some Ye supporters suggested that those complaints were from disgruntled Taylor Swift fans. Ms. Swift, with whom Ye has a long and well-documented feud, has yet to announce any China dates for her Eras tour. (Several Shanghai government advisers recently called on the city to loosen its concert approval processes, citing performers like Ms. Swift, who they said were like “walking G.D.P.”)

The anti-Ye comments have since disappeared from the government website.

Siyi Zhao and Claire Fu contributed research. Julia Jacobs contributed reporting.

After racing across Russian fields in an American Stryker armored fighting vehicle this month, the six-man Ukrainian assault team dismounted in a tree line about 700 yards from the enemy’s trenches and waited for the order to attack.

When it came, Afonya, a 40-year-old construction worker drafted into the Ukrainian military just two months ago, said the Ukrainian soldiers were met with a hail of gunfire as soon as they moved from their hastily dug foxholes. He was hit in the hand by a bullet that shattered a bone.

Three members of the assault team were injured and pulled back while the other three waited for reinforcements to resume the attack in the Kursk region of Russia.

“There were too many of them,” Afonya said in an interview at a hospital in eastern Ukraine, where he was recovering after being evacuated.

More than a month after Kyiv launched its incursion into Kursk — sweeping across nearly 500 square miles and capturing around 100 Russian towns and villages in a few short weeks — Russian resistance is stiffening, Ukrainian soldiers interviewed near the border with Russia said as they moved to and from the front last weekend.

President Volodymyr Zelensky told reporters on Thursday that Russian forces had begun a concerted counterattack in Kursk.

Some of the heaviest battles have been taking place on the western edge of the new front, according to the soldiers and combat footage geolocated by military analysts. But the battle lines remained fluid, and there was little reliable information about the scale of Russia’s operation or how successful it has been in reclaiming territory.

Ukrainian soldiers said that even as Russia was counterattacking in some locations, they remained on the offensive along other parts of the Kursk front. But their advances have slowed and clashes are growing deadlier as Moscow deploys reinforcements and increases its aerial bombardments.

“There is more resistance,” said Yurii, 21, who was with one of the first Ukrainian units to cross the border when the incursion was launched on Aug. 6. There are more drones, “more shells, and even anti-tank guided missiles,” he said. “Their intelligence is also working very hard. As soon as a vehicle moves out, their artillery starts firing immediately.”

The Ukrainian soldiers spoke on the condition that only their first names be used, in accordance with military protocol. They also asked that their brigades not be named out of concern that it could give the Russians insight into the location of their forces.

While some of the information they provided is corroborated by geolocated combat footage, details about specific offensive movements could not be independently verified.

As powerful Russian guided bombs thundered in the distance on a recent day and a puff of smoke overhead marked the spot where a Russian surveillance drone was shot out of the sky, one group of soldiers, speaking on the side of a road near the border, said that more Russian troops were joining the fight every day.

Dmytro, a 40-year-old member of a drone unit, said the fighting in Kursk was still less intense than other battles he has fought in over the course of the war, but that is changing.

“They’re trying, but so far, nothing’s working for them,” he said. “We are still attacking.”

The soldiers said that much of the fighting before the Russian counterattack was for small tactical advantages — like taking control of a ridge or hill — that could prove useful in future battles.

Ukraine is using new forward positions inside Russia to disrupt Moscow’s logistical operations, attacking roads and bridges along critical supply lines feeding Russian forces inside Ukraine, soldiers said.

And they are continuing efforts to isolate a large group of Russian soldiers in a 270-square-mile pocket of land between the Ukrainian border and the meandering Seym River about 10 miles inside Russia, they say.

Ukrainian forces destroyed all of the bridges across the river and are targeting temporary pontoon bridges as soon as they are spotted, according to soldiers, satellite imagery and geolocated combat footage. But the Russian counterattack this past week appeared designed to relieve pressure on that pocket of land.

Russia’s Defense Ministry said its country’s forces had “penetrated” into the Kursk region, capturing 10 settlements.

The Institute for the Study of War, whose analysts use geolocated combat footage to track daily battlefield developments, wrote on Friday that they had yet to observe visual confirmation to support the Kremlin’s claims, with Russian soldiers appearing to be in partial control of two villages.

At the same time, Ukrainian forces breached the border in a new location west of the original incursion, according to combat footage released by both sides and geolocated by military analysts. The state of the fighting there is unclear, but it could complicate Russia’s counterattack.

As they battle to hold onto their gains, the Ukrainian soldiers said the campaign was coming at a steep cost.

“Every centimeter of our advance costs human lives,” said Serhii, a 40-year-old Ukrainian soldier whose home village, Sumy, is close to the Russian border.

The Kremlin is clearly hoping the Ukrainian military has overextended itself, leaving outnumbered and outgunned Ukrainian forces in the country’s eastern Donbas region vulnerable as Russia continues to press the attack there, military analysts say.

President Vladimir V. Putin has sought to minimize the first invasion of Russia since World War II as a mere distraction. While saying it was a “sacred duty” to expel Ukrainian forces, he said Russia’s main priority remained seizing Ukrainian lands.

Mr. Zelensky and Ukraine’s top military commander, Gen. Oleksandr Syrsky, have said the offensive has multiple goals, including drawing Russian forces from other parts of the front.

While the Ukrainian leaders acknowledged that the Kremlin has resisted pulling its best forces from the hottest parts of the eastern front as they had hoped, General Syrsky has maintained that the Kursk offensive is still affecting Russia’s ability to sustain other operations around the battlefield as it moves some 60,000 soldiers to the Kursk front.

Bill Burns, the C.I.A. director, told a conference in London last weekend that the operation was “a significant tactical achievement” that had boosted Ukrainian morale and exposed Russia’s weaknesses.

Mr. Zelensky has said the offensive is a part of a “Victory Plan” that he will present to President Biden and the two candidates vying to replace him — Kamala Harris and Donald J. Trump — on a trip to the United States this month.

Whatever the ultimate plan, Kursk is now clearly another violent front in an already sprawling war.

“Same war, different place,” said Dmytro, the drone operator.

As the Russians fight to halt the Ukrainian advance, soldiers said, they are using the same tactics they use inside Ukraine — a scorched-earth approach that has left dozens of Ukrainian towns and cities in ruins.

But the Russian bombardments are now devastating Russian homes, soldiers said.

“I bring the guys food, fuel, diesel, and gasoline over there,” a 56 year-old soldier who works in logistics, also named Serhii, said at a rest stop on the Ukrainian side of the border. “It was strange, when our troops first came, everything was intact. The roads were fine, the warehouses were untouched.”

“But after a couple of weeks, everything was destroyed, shattered,” he said. “They are destroying their own villages.”

Those claims by the Ukrainian soldiers were supported by combat footage showing Russian strikes on Russian villages and towns occupied by Ukrainian soldiers.

In Sudzha, the largest town under Ukrainian control, some high-rise and administrative buildings were destroyed as the Ukrainians advanced, but independent Western journalists who visited the region in the first days after it fell noted that the level of destruction was minimal compared with places in eastern Ukraine seized by Russian forces.

That is fast changing.

“Now, when you stand on a hill, for example, and look at Sudzha, you wake up every morning thanking God that you’re alive,” said Serhii, the soldier from Sumy. The town, he said, is a “land on fire.”

As he prepared to head back into Russia at nightfall, he said his heart was heavy with emotion. He understood the mission and thinks it is important, but would rather not have to fight on foreign soil.

“It’s one thing to defend your own land, but another to be over there,” he said.

Liubov Sholudko contributed reporting from eastern Ukraine.

Paris threw its last Olympics party on Saturday, a buoyant, nostalgia-tinged celebration of the 2024 Games that drew tens of thousands of cheering spectators to the streets of the French capital for a parade of athletes and an outdoor concert around the Arc de Triomphe.

The festivities started with smoky blue, white and red fireworks, echoing the start of the opening ceremony on the Seine. Flag-waving crowds then roared and sang France’s national anthem as more than 300 French Olympic and Paralympic contestants paraded up the Champs-Élysées on a giant white runway.

“Thank you all,” Teddy Riner, the French judo legend, told ecstatic spectators as they sounded air horns and chanted athletes’ names. “It was incredible!”

Medal-winning athletes were later decorated with state honors, some of them by retired French sports legends, and a handful were honored by President Emmanuel Macron himself. France won 64 medals, putting it in the top five of the Olympics medals count. And it earned 75 medals at the Paralympics.

After night fell, the Olympic cauldron floated into the air one last time and a highlight reel of the Games was projected onto the Arc de Triomphe. Performers from the opening and closing ceremonies also returned for an encore on a ring-shaped stage around the famous monument (including Philippe Katerine, a.k.a blue Smurf guy). French fencers, rugby players and others led the concertgoers through giant karaoke sessions.

It was a day of summer revelry before the fall doldrums set in, the end of an Olympic bubble that enchanted France and allowed it to forget, for a time, its current political turmoil and a looming government budget crunch. It was also a final opportunity to watch the Phryges, the widely beloved mascots of the Paris Games, in all their googly-eyed glory.

“We rose to the challenge and it went very well,” said Marie-Laure Bordes, 48, peeking from the back of the crowd to catch a glimpse of the Champs-Élysées, where thousands of volunteers and staff members from the organizing committee also paraded to widespread applause. She cited the smooth organization and euphoric mood as her family’s highlights of the Games.

“And Léon Marchand!” her teenage son said, referring to the French swimming superstar,

“Now that it’s over,” Ms. Bordes added, “there is a feeling of nostalgia.”

Spectators said they rushed to snap up the free tickets to the event, largely to relive the unexpected unity and fervor that gripped Paris during the Games after months of hand-wringing about whether France would be ready — and enthusiastic enough — for the Games.

“France is a grumpy place,” said Maryline Bregeon, 63, a retiree who volunteered at the opening ceremony and at the horseback riding events at Versailles. But the mood shifted noticeably during the Games, she said on the sidelines of the parade, wearing her official turquoise volunteer jersey.

“People were talking to each other on public transit,” she marveled. “Usually that never happens.”

France’s leaders are keen to capitalize on that spirit.

Anne Hidalgo, the mayor of Paris, wants to keep the Olympic rings on the Eiffel Tower, at least until Los Angeles hosts the 2028 Summer Games, although her plans have run into opposition.

Mr. Macron wants to make Sept. 14 a national day to celebrate sports. Weakened politically after inconclusive snap elections and forced to appoint a conservative prime minister who is not from his party, the French president has tried to portray the Games as an example of much-needed national unity.

“Who could understand that we know how to reach out and surpass ourselves to make the Olympic and Paralympic Games a success, but that we can’t do the same to build France and respond to the urgent needs of the French people?” Mr. Macron told Le Parisien, a newspaper, on Friday.

Mustapha Belfodil, 33, an Algerian doctor who works in Grenoble, in the French Alps, said he had taken a week off to volunteer as a medic at the track and field events — and had no regrets.

“I wanted to relive these moments,” he said at the parade, citing the victories by Mr. Marchand and the triumph of Imane Khelif, the gold-medal-winning Algerian boxer, as among his favorites.

But in the end, he said, “there were too many to choose from.”

Prime Minister Justin Trudeau’s Liberal Party should be a shoo-in for a parliamentary seat at the southern point of the island of Montreal.

The district has been a stronghold for his party for more than half a century. It was home to another Liberal prime minister a generation ago. The base for a former Liberal justice minister. An easy drive to Mr. Trudeau’s own redoubt in the city.

And yet, days before a special election on Monday to choose the district’s member of Canada’s Parliament, polls show a tight three-way contest. For many lifelong Liberals, the problem is clear: It is Mr. Trudeau himself.

“I am a Liberal supporter, but it’s almost like enough is enough,” Michael Altimas, 79, a retired city bus driver, said during a walk on a sunny day along the district’s long pedestrian commercial street. “For the most part, he’s been a good prime minister.

“But he’s had nine years,” Mr. Altimas added, “and people are hearing often enough that he messed up, and they don’t want to support him anymore.”

The election to fill a vacancy in the district has become a referendum on Mr. Trudeau, the once golden boy of Western leaders who is now fighting for his political survival. His own Liberal Party members are increasingly calling for him to step aside, worried that the party risks a drubbing in the next general election under the deeply unpopular leader.

Public grumblings about his leadership grew louder over the summer after his party lost a special election in Toronto in June — in another stronghold — and after President Biden’s decision to step down as the Democratic candidate in the U.S. election suggested a path forward for Canada’s Liberals.

The stunning loss in Toronto has raised the stakes for the election in Montreal. Underscoring Mr. Trudeau’s cratering popularity is his near-total absence from the local campaign.

His face does not appear on posters in the Liberal campaign office’s storefront windows or on the district’s lampposts. He appeared once to introduce the Liberal candidate but hasn’t been back since. Other party leaders, by contrast, have been visibly present.

“Right now, Justin Trudeau has no political coattails,” said Nik Nanos, a leading pollster in Canada. “He’s become the lightning rod for the general disaffection directed at the Liberal Party.”

When Mr. Trudeau was first elected in 2015, he enjoyed “one of the strongest brands in the polling history of Canada,” Mr. Nanos said. But Mr. Trudeau’s approval ratings are now stuck just above 20 percent and trail by double digits those of the main opposition Conservative Party leader, Pierre Poilievre.

For the past year, Mr. Poilievre has set the national political agenda through relentless attacks on Mr. Trudeau’s handling of issues most concerning to voters, including the economy, housing and immigration. The Liberal government has often reacted to Mr. Poilievre’s criticisms with hurried policy tweaks.

Mr. Trudeau has vowed to run for a fourth term in the next general election, which must be called by the fall of 2025. But a string of recent developments has amplified the pessimism of Liberals and former allies: the abrupt resignation of the Liberal Party’s campaign director; the labor minister’s resignation and the transportation minister’s public angling for a provincial post; an exodus of senior government employees; and the sudden decision by the New Democratic Party to abandon an agreement to support Mr. Trudeau’s party for fear of being tainted by the association.

A loss in Montreal’s special election could embolden internal opposition and further undermine Mr. Trudeau’s public image.

The district, called LaSalle–Émard–Verdun, has been redrawn a few times but has been a Liberal fortress since the 1960s. Its traditionally working-class and immigrant residents backed the Liberals, as have newcomers to its gentrifying neighborhoods.

In the last general election in 2021, the Liberal candidate won by 20 percentage points over his closest rival. Today, polls show the Liberals with a slight edge but locked in a tight battle against two opposition parties, the Bloc Québécois, a national party that supports Quebec independence, and the New Democratic Party.

Many voters interviewed on Wellington Street in Verdun — a long pedestrian street filled with restaurants, cafes and neighborhood stores — singled out Mr. Trudeau as influencing their decision.

“It’s hard to imagine a world in which Trudeau gets re-elected,” said Christopher Gaudreault, 28, a classical pianist, who has voted for Liberals, Greens and New Democrats. “What I’ve been hearing in my circles is that pretty much everyone across the board is fed up with Trudeau for various reasons.

“People are just eager for a change and hope for something better,” he added.

So far, Mr. Trudeau has wielded the extraordinary powers conferred on Canada’s political party leaders to quash internal dissent.

Last November, Percy Downe, a senator who served as chief of staff to a former prime minister, became one of the first Liberals to suggest publicly that Mr. Trudeau step aside for a fresh face before the next election. Few Liberals followed suit — at least publicly.

Mr. Downe, in an interview, explained that most senators — who, under Canada’s Constitution, are appointed rather than elected — have been named by Mr. Trudeau. At the same time, the more powerful members of the House of Commons fear questioning Mr. Trudeau, who, like all Canadian party leaders, enjoys near-total control over individual party members’ electoral prospects.

“You won’t be allowed to run in the next election,” Mr. Downe said, pointing out that no candidate can run in a district without the party leader’s endorsement. He called party leaders’ absolute power over their members “a fundamental weakness in our democratic system.”

After the Liberals’ loss in the Toronto race in June, the Canadian news media was filled with anonymous Liberal criticism of Mr. Trudeau’s leadership. Only one Liberal member of Parliament — who has announced that he is retiring from politics — openly called on Mr. Trudeau to step down. Other Liberal lawmakers called for an emergency national meeting to discuss the party’s future.

Mr. Trudeau brushed away those calls.

Royce Koop, a political scientist at the University of Manitoba, said Mr. Trudeau had succeeded because time was running out for the Liberals to change leaders before the next election.

“If you’re Trudeau and you’re trying to hang on, delay is a good tactic,” Mr. Koop said.

Even though Mr. Trudeau has largely stayed away from the LaSalle–Émard–Verdun race, his grip on the party is still evident. Mr. Trudeau handpicked the candidate, a city councilor named Laura Palestini, in mid-July — angering three other Liberal candidates vying for the candidacy.

One of them, Christopher Baenninger, an entrepreneur, said Liberal Party officials had reassured him that the candidate would be elected by members in an open race. He said he had spent five months gathering support, knocking on doors seven days a week.

Among Liberal supporters, he said half were committed “no matter what.” But he said the other half were “tired Liberals, who were, like, ‘Trudeau’s been in power for nine years now. We’re looking for something fresh.’”

The open nomination had made Liberals “feel like their voices are heard,” said another former candidate, Eddy Kara, a Liberal organizer and a filmmaker. But the last-minute decision to close the nomination and parachute in a candidate risks leaving Liberals feeling disenfranchised and “exacerbating people’s negative perceptions” about politics, he added.

Parker Lund, a Liberal Party spokesman, said in an email that the selection of Ms. Palestini was “fully in line with our national nomination rules.” He did not respond to requests to interview a senior party official about the state of the Liberal Party.

At Ms. Palestini’s campaign office, the campaign manager, Marie-Pascale Des Rosiers, said the candidate was not granting interviews and declined to let a journalist accompany her while campaigning.

A few doors away, at the New Democrats’ campaign office, the excitement about a possible upset victory was palpable. The party’s leader, Jagmeet Singh, whose own electoral district is in a Vancouver suburb, has visited Montreal about a dozen times to campaign with the party’s local candidate, Craig Sauvé, a city councilor.

The Bloc Québécois, whose candidate is Louis-Philippe Sauvé, has also expressed optimism about winning.

The New Democrats’ candidate, Mr. Sauvé, said he was knocking on doors two to three times a day.

“There is a generalized fatigue,” he said, “with regards to the Liberal Party.”