Brazilian investigators on Saturday began analyzing the black boxes from a São Paulo-bound flight to try to understand why the passenger plane fell from 17,000 feet on Friday, in a crash that killed all 62 on board.

But to aviation experts around the world who watched the videos showing the 89-foot plane spinning slowly as it plummeted before crashing almost directly on its belly, the question of what had happened was simple to answer: The plane had stalled.

In other words, the plane’s wings had lost the lift needed to keep the aircraft aloft, causing it to stop flying and start falling.

“You can’t get into a spin without stalling,” said John Cox, an airline pilot for 25 years who now aids plane crash investigations. “It’s A plus B equals C.”

The question of why VoePass Flight 2283 might have stalled, however, remained a mystery.

Did it lose significant speed? Did its nose pitch up too high? Did ice build up on its wings? Did an engine fail? Was its stall-warning system working? Were the two pilots tired or distracted?

“The main thing we know is that it’s never one thing,” said Thomas Anthony, director of the aviation safety program at the University of Southern California.

The plane was carrying 58 passengers and four crew members on the nearly two-hour scheduled flight from Cascavel, Brazil, to São Paulo on Friday when it crashed in a gated community in the small city of Vinhedo, shortly before reaching its destination. No one on the ground was injured.

Crash investigators in Brazil said on Saturday that they had recovered the plane’s two black boxes — one containing flight data and the other recordings from the cockpit — and were working to extract information from them.

“There is still no estimated completion date for this work,” Marcelo Moreno, Brazil’s chief crash investigator, said. “We are prioritizing quality over speed.”

The leading crash theory so far is that the plane may have stalled partly because it suffered from severe icing, meaning ice formed on its wings or on other parts of the plane, reducing its aerodynamic abilities and increasing its weight. With such icing, a plane has to travel at faster speeds to avoid stalling, experts said.

“The way the aircraft fell, spinning out of control, is characteristic of someone who lost the functionality of the wing and the aircraft controls,” said Celso Faria de Souza, a Brazilian aeronautical engineer and forensic expert in plane crashes. “This can happen because of ice.”

Brazilian officials had issued a warning about the potential for severe icing where the plane was flying when it crashed. And shortly before the disaster, a different passenger plane had experienced such icing nearby, the pilot of that plane told the Brazilian news channel Globo.

Passenger planes have systems to break up ice that forms on the wings. On the plane that crashed — an ATR 72-500 turboprop built in 2010 — that system consisted of rubber tubes on the wings that are supposed to inflate and deflate to break up any ice.

“Did the crew activate the anti-icing system?” asked Jeff Guzzetti, a former crash investigator with the Federal Aviation Administration. “Or did they activate it and it failed?”

Icing was a main cause of a 1994 American Eagle crash of the same ATR plane model in Indiana, but the manufacturer has since improved the de-icing system.

Mr. Cox, the pilot and crash investigator, said that publicly available flight data suggested that the plane was traveling roughly 325 miles per hour when its speed dropped sharply in the minutes before the crash. The speed did not drop far enough to cause a stall, he said, unless icing was extremely severe.

“If there is enough ice, then it changes the shape of the wing, and that could cause it to stall at a much higher speed,” he said.

However, the experts said, icing alone should not lead to a crash. Under most circumstances, pilots can prevent icing from causing a stall, even if the plane’s systems fail.

If the system that warns of icing malfunctioned, the pilots should have been able to see the ice on the wings and windshield wipers, they said. And if the system to break up the ice malfunctioned, pilots could have lowered the plane’s altitude, where warmer air would have melted the ice. “We’re talking about Brazil here, not Antarctica,” Mr. Cox said. The temperature on the ground where the plane crashed was around 63 degrees.

To descend to a lower altitude, pilots typically first alert air traffic controllers. But Brazilian officials said the pilots did not communicate with controllers just before the crash. “There was at no time a declaration of any type of emergency from the aircraft,” Mr. Moreno, the crash investigator, said.

The experts said they were mystified by the lack of communication.

“They may have tried to speak and the radio failed, the communication failed,” said Joselito Paulo, president of the Brazilian Aviation Security Association. “Or they made the communication, but it wasn’t intercepted by air traffic control.”

“If there was no communication,” he added, “it was something very quick, unexpected.”

Marcel Moura, operations director of VoePass, the airline operating the flight, told reporters that investigators would look at all possible causes.

“The plane is sensitive to ice. It’s a starting point,” he said. “But it’s still very early to make a diagnosis.”

At the crash scene inside the gated community on Saturday, officials picked through the wreckage to try to extract the bodies of victims. By Saturday evening, they had found 42 of the 62 people who died.

Among the passengers on the flight were at least four doctors on their way to a medical conference, as well as university professors, a D.J., a bodybuilder and a judo referee, according to local news reports. All of the passengers were Brazilian, although three held dual citizenship with Venezuela and one with Portugal.

An Israeli airstrike early Saturday hit a school compound in northern Gaza where displaced Palestinians were sheltering, killing dozens of people, according to Gazan officials.

The Israeli military acknowledged the attack, but said Hamas and another armed Palestinian group were using the facility for military operations and attacks on Israel.

The strike in Gaza City, the latest in a string of attacks on schools turned into shelters, drew strong condemnation from the European Union and the United Nations, with Josep Borrell Fontelles, the top E.U. diplomat, saying, “There’s no justification for these massacres.”

The strikes have taken place alongside mounting international pressure on Israel to conclude a deal for a cease-fire and an exchange of hostages held in Gaza and Palestinian detainees, with President Biden and the leaders of Egypt and Qatar saying this week that “the time has come.”

The Gaza Civil Defense emergency service said more than 90 people were killed. But that number could not be confirmed, and two doctors at a hospital in the area gave slightly lower totals. Gaza health officials do not distinguish between civilians and combatants when reporting casualties.

The Israeli military did not provide a death toll. But it questioned the Gaza authorities’ statements, saying that its own assessment of the incident was at odds with the reported death toll.

Daniel Hagari, the military’s chief spokesman, said that intelligence gathered by Israel indicated that “no women and children were present” in the building struck by Israeli forces. Regarding the casualty count, he said: “These figures do not distinguish between combatants and noncombatants, and they do not match the information” held by the Israeli military.

The Civil Defense emergency service in Gaza said that the strike hit as more than 200 people gathered before sunrise in a prayer hall to worship. More than 2,000 displaced people had been staying at the shelter, the Al-Tabaeen school in the Al-Daraj neighborhood, the emergency service said.

The attack was launched as the United States, Egypt and Qatar mounted a new effort to restart negotiations for a truce in Gaza next week, with fears rising of an escalation in the conflict between Israel and Iran. Analysts said President Biden and his allies hope the prospect of a cease-fire in Gaza will dampen tensions between Iran, its militant allies and Israel, and put brakes on the momentum toward a regional war.

For days, Israel has tensely awaited retaliation for the assassination of top leaders from Hamas and Hezbollah, both groups backed by Iran. President Biden and the leaders of Egypt and Qatar called Thursday for more talks between Israel and Hamas to end the war in Gaza, saying they would be willing to present a “final bridging proposal” to both sides.

There is “no further time to waste,” the leaders said in a joint statement, a sign of the growing impatience over the stalled peace talks. Hours later, Prime Minister Benjamin Netanyahu of Israel said he would send negotiators to talks next Thursday, while Hamas has yet to respond to the offer.

In Gaza, however, Israel’s campaign — now in its 11th month — dragged on.

At least 17 school buildings have been targeted over the last month, killing at least 163 Palestinians, according to the United Nations. The Israeli military said Hamas has cynically exploited the compounds, and the displaced people sheltering inside, for its own protection; international law experts say Israel needs to do more to avoid harming civilians.

The Israeli military defended Saturday’s attack, saying in a statement that “the strike was carried out using three precise munitions,” and that several steps were taken to mitigate civilian casualties, “including the use of a small warhead, aerial surveillance and intelligence information.” At least 19 militants from the Hamas and Islamic Jihad armed groups were killed in the attack, the statement said.

“The compound, and the mosque that was struck within it, served as an active Hamas and Islamic Jihad military facility,” said Lt. Col. Nadav Shoshani, an Israeli military spokesman, without providing details. He added that the information released by the Gaza authorities in the past has “proven to be sorely unreliable.”

The Gaza Health Ministry’s numbers are believed to be broadly reliable, though there is often uncertainty in the immediate aftermath of specific strikes, and the destruction of the territory’s health system has made tolls harder to track.

Saturday’s airstrike hit two floors, one of which was used for communal prayers and the other for sheltering women and children, Mr. Basal said. He said the prayer hall inside the school complex has been used for worship since the beginning of the 10-month war.

The Civil Defense spokesman, Mahmoud Basal, said 11 children and six women were among those killed in the strike on Saturday, adding that many people were seriously wounded.

Many of those wounded in the Israeli strike, including children, were arriving with severe burns covering much of their bodies, said Tayseer al-Tanna, a surgeon at Al-Ahli Hospital in Gaza City, who called the scene “very difficult to watch.”

Dr. al-Tanna said doctors had been forced to amputate several badly burned and mangled limbs since early morning. They had almost no painkillers to give their patients, he said, and did not have enough time to fully sterilize the already overused equipment between operations.

Fadel Naim, a medical official at Al-Ahli Hospital who served for years as dean of the medical college at the Islamic University of Gaza, widely seen as a Hamas stronghold, said the hospital had received at least 70 bodies since Saturday morning. The strike was followed by a flood of people searching for loved ones missing in the wake of the explosion, he said.

Khamis Elessi, a doctor at the same hospital, in Gaza City, said more than 73 identified bodies were brought to the hospital morgue, along with another 10 who have yet to be identified because they were disfigured in the explosion.

Israeli forces have recently been scaling up military attacks throughout Gaza in areas where they had previously fought Hamas, saying the fighters had regrouped. Hamas has fought a dogged insurgency, resisting Israeli attempts to quash the group. That has raised concerns in Israel that a decisive military victory may be out of reach — even as Gazan civilians continue to face dire conditions.

Troops had previously moved in on the Al-Daraj neighborhood in early July as part of a renewed ground offensive in Gaza City. But the Israeli military appears to have wound down its ground operation there, even as it continued to conduct airstrikes in the area.

The U.N. and other rights organizations have repeatedly said that there is no safe place in Gaza, as areas people are ordered to evacuate to are subsequently targeted by Israeli airstrikes. Almost the entire population of Gaza — more than two million Palestinians — has been displaced, many people multiple times.

Philippe Lazzarini, the head of the U.N. agency for Palestinian refugees, called Saturday’s deadly attack “another day of horror” in Gaza. He called on all sides not to harm civilians or use schools for military purposes.

“It’s time for these horrors unfolding under our watch to end,” he said on social media. “We cannot let the unbearable become a new norm.”

Ameera Harouda and Rawan Sheikh Ahmad contributed reporting.

Russia is pushing back against Ukraine’s largest assault into Russian territory since the start of the war, sending troop reinforcements, establishing strict security measures in border areas and conducting airstrikes, according to the Russian Defense Ministry. One of the strikes on Ukrainian troops involved a thermobaric missile that causes a blast wave and suffocates those in its path, the ministry said.

But even as Russia has halted the quick advances made by Ukrainian troops who launched a surprise cross-border attack five days ago into the southwestern region of Kursk, Ukrainian forces seem to be holding ground. They claimed on Saturday to have captured a small village in the neighboring Belgorod region, and analysts say their forces control most of the Kursk town of Sudzha, about six miles from the border.

Pasi Paroinen, an analyst from the Black Bird Group, a Finland-based organization that analyzes satellite imagery and social media content from the battlefield, said in an interview that evidence suggested that Moscow had been able to stall the major advances in Russian territory late in the week.

“We’re now entering the phase where the easy gains have been made,” he said of Ukraine’s initial advance. “This phase, for the first three days, saw the most rapid movement,” he added. “And yesterday, I think, we started to see the effects of the Russian response.”

What all of this means for Ukraine is not yet clear. In the third year of a war that has seemed largely frozen along a 600-mile front line in eastern and southern Ukraine, the decision by Ukrainian forces to cross the border into Russia apparently surprised not just Russia, but also the United States, other Western partners and analysts who spend their days following the war’s troop movements.

Some have speculated that Ukraine hopes to draw Russian troops away from the front lines in Ukraine, giving battle-weary Ukrainian troops a needed rest, although analysts say that has not happened.

But for Russia, the fallout from this past week’s incursion has been a public black eye for President Vladimir V. Putin.

Andrei Gurulyov, a retired military officer who is now a member of Parliament, on Friday condemned Russia’s response and level of preparedness.

“There is no military system in place for guarding the state border, no reserves and no second lines of defense,” he said on Telegram, adding, “If the Ukrainian Armed Forces spent two months preparing for this, how did we miss it?”

Dara Massicot, a senior fellow at the Carnegie Endowment for International Peace in Washington, said that the Ukrainian incursion represented for Russia “a system failure on multiple levels — intelligence, defense and emergency response.”

“Border regions were placed on an emergency footing in 2022 to be specifically postured for these kinds of contingencies,” Ms. Massicot said, “which makes the lapse even more glaring.”

To counter the incursion, Russia’s military appears to be relying mostly on units that were already deployed near the area, according to an analysis by the Institute for the Study of War, a Washington-based think tank. Most of those units consist of military conscripts and irregular forces, as opposed to the battle-hardened soldiers fighting in Ukraine.

Russia’s Defense Ministry said on Friday that it was “continuing to repel” Ukraine’s military. And Alexander Kharchenko, a military correspondent for RIA Novosti, a Russian state-run news agency, said on Saturday on Telegram that while Ukrainian troops had easily entered populated areas in the region days earlier, “now they are met by a dense barrier in front of every village.”

Russian officials and the state news media have repeatedly claimed in recent days to have the situation under control, however, only to then lose more ground.

The new offensive has alarmed the Russian public, and if conscripts are killed, that could bring the war home in a way that losses along the front lines in Ukraine have not.

The conflict also threatens to spill over into neighboring Belarus: On Saturday, Belarus, a Russian ally, said it was sending more troops to protect its border with Ukraine, claiming that Ukrainian drones had violated its airspace during the incursion into Russia. Belarus’s defense minister said on Telegram that its air defense forces had destroyed several Ukrainian drones that violated Belarusian airspace on Friday.

Ukraine has managed to keep much of its troop movements hidden, telling soldiers not to say anything publicly or post any videos. But there have been exceptions, including when a group of soldiers posed in a video outside a facility run by Gazprom, the Russian state-owned gas monopoly, on the outskirts of Sudzha; the video was verified by The New York Times.

Satellite imagery showed several damaged or destroyed buildings in the center of Sudzha, including the office of the public prosecutor and what appears to be an apartment building. The Times reviewed additional satellite imagery that showed fighting continuing as of Friday, with visible smoke plumes and a fire.

Mr. Paroinen said that a few Russian troops were likely still holding out in the center of Sudzha, and that the Ukrainians were trying to set up defenses east of there. But he also said that Ukrainians had encountered stiff resistance when they tried several times to take control of Korenevo, a town about 30 miles to the northwest.

“This may have caused quite a bit of trouble for the Ukrainians,” he said.

Since launching the cross-border attack on Tuesday, Ukrainian officials have said little publicly about it, aside from pointing out that the country’s allies have supported the offensive, at least tacitly.

On Saturday, President Volodymyr Zelensky of Ukraine went further, offering his biggest acknowledgment of the incursion into Russia so far. He referred to the fact that Ukraine had pushed the war “out into our aggressor’s territory” and thanked the military for making it happen.

A day before, Mr. Zelensky obliquely alluded to the offensive and cited one benefit from it: newly captured Russian troops who could be traded for Ukrainian prisoners of war. He expressed gratitude to Ukrainian troops “who are replenishing the ‘exchange fund.’”

“This is extremely important and has been particularly effective over the past three days,” Mr. Zelensky said. “We must return freedom to all our people who remain in Russian captivity.”

Since Russia invaded Ukraine in February 2022, Kyiv’s allies have been wary of any Ukrainian incursions onto Russian soil, fearing that they could provoke Moscow and escalate the war. Matthew Miller, a spokesman for the U.S. State Department, suggested this past week that the Ukrainian incursion did not violate American guidance.

It is not clear whether Russia will redeploy better-equipped forces from the front lines inside Ukraine to suppress Kyiv’s foray into Russian soil. That would take time. It would also relieve pressure on Ukrainian troops who are outnumbered and have been losing ground in eastern and southern Ukraine.

So far, Russian forces have not let up in their frontline pushes. In recent days, they have made some inroads near the embattled eastern Ukrainian towns of Chasiv Yar, Toretsk and Pokrovsk, the Institute for the Study of War said. On Friday, the British Defense Ministry said that Russian troops in the Donetsk region of eastern Ukraine had gained ground, pushing to about 10 miles from Pokrovsk.

After steadily losing ground this year, Ukraine’s success with this latest offensive has buoyed the country’s morale, at least if social media posts are any guide. The head of Ukraine’s Postal Service joked on Telegram that he planned to open a post office in Sudzha.

On Friday, the situation remained tense along the border, with both sides trying to evacuate residents. Russia declared states of enhanced security in the regions of Kursk, Belgorod and Bryansk. It said that would include counterterrorism operations, which allow for restrictive military and security actions inside Russia, as applied to parts of Chechnya for years during the war there.

The Ukrainian authorities, perhaps preparing for retaliation, said on Friday that they were evacuating 20,000 people from the Sumy region, across the border from Kursk.

Reporting was contributed by Anastasia Kuznietsova, Evelina Riabenko, Alina Lobzina and Christoph Koettl, Eric Schmitt and Constant Méheut.

Hundreds of people gathered several days ago outside a detention center known as “Zone 7” in Caracas, the Venezuelan capital, huddled around lists of prisoners, as they clutched plastic bags filled with meals they had packed for the inmates inside.

Eager for information about their detained loved ones, many told remarkably similar stories of sons, daughters and siblings arrested riding motorbikes, walking home from work, coming out of a bakery or stopping by a relative’s house in the days following Venezuela’s disputed presidential election.

They described arrests both sweeping and selective. And no one had been told what criminal charge their relatives faced.

The Venezuelan government has mounted a furious campaign against anyone challenging the declared results of the vote, unleashing a wave of repression that human rights groups say is unlike anything the country has seen in recent decades.

“I have been documenting human rights violations in Venezuela for many years and have seen patterns of repression before,” said Carolina Jiménez Sandoval, president of the Washington Office on Latin America, an advocacy and research organization. “I don’t think I have ever seen this ferocity.”

The country’s autocratic president, Nicolás Maduro, claimed victory in the July 28 election, but the government has yet to provide any vote tallies to support the announcement. The opposition, on the other hand, released tallies showing that its candidate had won in a landslide.

Now, experts say, Mr. Maduro, having seemingly been repudiated by a majority of his constituents, is bent on punishing those he considers disloyal.

“My son grabbed a flag and participated in a protest, but I don’t think he will be convicted for that, will he?” said María Vázquez, 62, a Caracas street vendor who supports the government and urged her son not to protest. “It’s worrisome.”

The Venezuelan government says it has arrested more than 2,000 people for participating in protests disputing the election results.

People were taken both in indiscriminate roundups, amid the protests, and later from their homes in targeted arrests, as the government launched what it called “Operation Knock-Knock,” according to interviews with family members and human rights activists documenting the detentions.

The surge in detentions is particularly alarming, rights groups say, because some arrests came after the president urged his supporters to snitch on their neighbors, using a government app that was supposed to be used to report issues like downed power lines.

“Maximum punishment! Justice!” Mr. Maduro said at a rally last Saturday. “There will be no forgiveness this time!”

The result has been an aggressive crackdown on dissent designed to silence anyone who dares question the election results, human rights activists said.

At least two human rights lawyers are in jail, including one who was arrested when he went to inquire about other detainees. Another activist was taken from the Caracas airport when she tried to leave the country.

When the authorities showed up at the home of María Oropeza, an opposition party leader in Portuguesa, southwest of Caracas, she live-streamed it. “I think first you should show me whether you have a search warrant, no?” she could be heard telling a police officer. “Because this is my home, private property.”

Jordan Sifuentes, the mayor of Mejía, the only opposition mayor in the state of Sucre, in northeast Venezuela, has been held for a week on unknown charges. Mayor José Mosquera of Lagunillas, in Zulia State, was held for six days after being accused of posting a tweet against the government, which he had denied.

Human rights activists and journalists learned in recent days that the government had annulled their passports, effectively trapping them in Venezuela.

People are leaving their houses without their phones, fearing that the authorities will stop them on the street and look at their messages for objectionable content. One man in Zulia was arrested after the police found a meme critical of the elections on his phone, his family said.

“It’s difficult to express in words the intensity and the indiscriminate nature of this wave of arrests,” said Gonzalo Himiob, vice president of the Penal Forum, a human rights organization tracking arrests since the elections.

Though the government claims that more than 2,000 people are in custody, Mr. Himiob said human rights organizations have only been able to document nearly 1,300 people detained.

“Maduro speaks of 2,000 detained, but it doesn’t seem true,” he said. “It seems more like an instruction. He wants to reach that number.”

On July 28, Mr. Maduro faced off against a little-known diplomat named Edmundo González, a stand-in for a more popular opposition leader, María Corina Machado, who had been disqualified by the government from running for office.

About six hours after the polls closed, the elections council announced Mr. Maduro had won another six-year term. Nearly two weeks later, the government has yet to publish any precinct-level elections data proving it.

The tallies collected by opposition observers on election night show Mr. González won by millions of votes.

Spontaneous protests erupted the day after the race, some of which led to clashes between demonstrators, security forces and armed civilian groups that support the government. At least two dozen people were killed, according to human rights groups. Hundreds were arrested.

But arrests continued days after the protests, sometimes on the word of anonymous informers who reported them on VenApp, an app that the government had originally introduced to report public nuisances.

The app has been removed from Google Play and the App Store, but is still available for those who have already downloaded it, according to Amnesty International.

Using civilian supporters to inform on neighbors has echoes of what has happened in Cuba, where the Communist government has long deployed an extensive network of community-based informants.

“Operation Knock-Knock is just beginning,” Douglas Rico, the head of Venezuela’s criminal investigations unit, posted on Instagram. “Report if you have been the target of a physical or virtual hate campaign through social media.”

The government appeared to be employing a “pluralistic” approach to quash dissent, said Ms. Jiménez of the research organization, using all the methods at its disposal, including technology, security forces, intelligence services, armed civilians and the armed forces.

“The array of tools the government is using,” she said, “is something not seen in previous cycles of repression in the country.”

Mr. Maduro insisted that the detained people had participated in an extreme-right fascist plot to oust him. People were paid to burn electoral centers and knock down statues of former President Hugo Chávez, he said, adding that they had confessed to their crimes.

The people who have been arrested will be charged with inciting hatred and terrorism, the government said, and activists said they had been referred to a specialized terrorism court in Caracas. Some of those arrested were caught committing acts of vandalism such as taking down government statues, but many others were merely in the wrong place at the wrong time, civil rights lawyers said.

The attorney general’s office did not respond to a request for comment.

On Thursday, the family of the opposition party leader Américo De Grazia, 64, announced on Instagram that he had been missing for more than 24 hours.

His daughter, María De Grazia, 30, said that after receiving threats on social media, her father, a former mayor and congressman, left his house in Upata and traveled 450 miles to Caracas. After five days there, he suddenly disappeared.

The family learned he was in jail, but said they were not told why.

“They did not come to the house with an arrest warrant,” said Ms. De Grazia, who lives in exile in Houston. “If a family member had not gone looking under rocks for two days, we would still not know where he was.”

The government was clinging to power, she said, by arresting everyone from student leaders to well-known politicians to ordinary citizens. Opposition activists barely stood a chance against such an organized apparatus.

“We are going to war armed with a plastic fork,” she said.

Nayrobis Rodríguez contributed reporting from Sucre, Venezuela, and Sheyla Urdaneta from Maracaibo, Venezuela.

Ukrainian forces pressed deeper into Russia on Friday, trying to capitalize on their surprise cross-border offensive, as Moscow moved quickly to shore up its defenses against the largest assault on Russian soil since the war began.

After capturing several small settlements the last few days, Ukraine was battling to take full control of a town near the border and sending small units to conduct raids farther into the southwestern Russian region of Kursk.

At the same time, the Russian military announced it was sending more troops and armored vehicles to try to repel the attack. Russian television released videos of columns of military trucks carrying artillery pieces, heavy machine guns and tanks.

Perhaps preparing for retaliation, Ukrainian authorities on Friday said they were evacuating 20,000 people from the Sumy region, which sits across the border from Kursk.

The flurry of movement underscored the extent to which Ukraine’s attack had introduced an unpredictable new element into a war that has been progressing in slow motion elsewhere on the battlefront. Ukraine’s success so far could provide a much-needed boost to a country whose forces have been steadily losing ground for many months, while bringing the war home to Russian civilians in a more serious way.

But military analysts have questioned whether the operation is worth the risk, given that Ukrainian forces are already stretched. It is also not clear whether the mission will help Ukraine improve its position on the rest of the battlefield by forcing Russia to divert troops from elsewhere to reinforce the border region.

The Ukrainian military has enforced a policy of silence about the operation, and it has not publicly acknowledged launching a cross-border attack.

Kyiv’s allies in the past have been wary of Ukrainian incursions in Russia, fearing that it could escalate the war, but there have been no public indications from Western capitals that they oppose the assault. The United States has said that the Ukrainian incursion does not violate American guidance.

However, senior American officials have said privately that they did not get a heads-up about the operation and were still seeking clarity about its logic and rationale.

The officials said they understood Kyiv’s need to change the optics and the narrative of the war, but that they were skeptical that Ukraine could hold the territory long enough to force Russia to divert significant forces from the offensives it is pressing in eastern and southern Ukraine.

“It’s a gamble,” said one senior administration official.

Still, Mykhailo Podolyak, a top presidential adviser, was upbeat about the international response. “Most quietly approve,” he wrote on social media on Thursday evening, adding that a significant part of the world now considers Russia “a legitimate target for any operations and types of weapons.”

The fighting in Russia showed no signs of abating on Friday, with the Ukrainian military saying that it had struck a Russian airfield in the Lipetsk region, which borders Kursk, hitting warehouses that contain guided aerial bombs. Local Russian authorities said a large drone attack had caused several explosions and that a fire had broken out at a military airfield.

The Ukrainian authorities also said a Russian strike on a supermarket in Kostiantynivka, an eastern Ukrainian town 200 miles south of the area of the fighting, killed 14 people and wounded 43 others. The claims from both sides could not be independently verified.

Military analysts said the attack across the border had involved elements of at least four brigades in a rare example of successful maneuver operations involving support from artillery, air defenses and electronic warfare, resulting in quick advances on the ground.

“It seems to be a fairly well-coordinated and planned combined armed operation,” said Franz-Stefan Gady, a Vienna-based military analyst. “You have electronic warfare assets that were deployed to jam Russian command and control. You have air defenses that were moved in to create air defense bubbles around the Ukrainian advance. And then you have fairly effective mechanized formations moving forward at a fairly steady pace.”

Mr. Gady and other experts said the main question now is whether Ukraine can maintain the momentum and turn the success on Russian territory into useful gains. The Ukrainian Army has few reserves it can pour into the fight, and it continues to suffer from shortages of weapons and ammunition, analysts say.

It also remains unclear what Ukraine ultimately hopes to accomplish. A senior Ukrainian official who spoke on the condition of anonymity to discuss the operation said the goal was to draw Russian troops away from other parts of the front line where Ukrainian units are struggling. But military experts said that Russia would likely be able to respond with reserves who were not fighting in Ukraine.

“Does it really solve any of the larger military strategic problems that the other parts of the front line are suffering from?” Mr. Gady asked.

Mick Ryan, a retired Australian Army major general and a fellow at the Lowy Institute, a research group, said one objective may be to boost morale in the Ukrainian population. “Given the past eight months of defensive operations, constant aerial attacks on infrastructure and ongoing power shortages, the will of the people will be at the forefront of the Ukrainian government’s considerations about the trajectory of the war,” he said.

A map of the battlefield by the Black Bird Group, a Finland-based organization that analyzes images from the battlefield, shows that Ukrainian troops have gained about 100 square miles of Russian territory since the beginning of the attack, although it remains unclear whether they have secured control of all of it. They have advanced past two lines of Russian defenses.

In particular, the Ukrainian Army has entered Sudzha, a small town of about 6,000 people six miles from the Ukrainian-Russian border. On Friday, Ukrainian troops claimed in a video that the town was under their control. The claim could not be independently verified.

Emil Kastehelmi, an analyst from the Black Bird Group, wrote on social media that some Ukrainian units appeared to be conducting probing raids further north in the direction of Lgov, a town about 50 miles from the border, in what appears to be a test of Russian defenses.

A video posted on social media on Friday morning and verified by The New York Times showed a column of destroyed Russian military vehicles just east of Rylsk, a Russian town west of the border area captured by Ukraine.

It remains to be seen whether Ukraine will try to push further into Russian territory to solidify control over the area it has captured, or retreat after a few days, as has happened in previous, smaller-scale cross-border raids.

Mr. Kastehelmi said Ukraine could not continue further north without widening its flanks and exposing itself to Russian counterattacks. “Time is also running against Ukrainians,” he wrote. “Russians won’t be disorganized forever.”

Eric Schmitt contributed reporting from Washington, D.C.