KYIV — Ukrainian troops battled for a third day Thursday in Russia’s Kursk region, occupying villages and part of a town, in what has become the Western-backed military’s largest cross-border incursion since the Kremlin’s invasion in 2022.

       U.S., Ukrainian and Russian officials all acknowledged the ongoing attack, which stunned Moscow and appeared to involve the use of armored fighting vehicles donated to Kyiv by the United States and its European partners — a development that drew no immediate objection from the Biden administration despite its past restrictions on such use of American weaponry.

       The surprise assault on Kursk, about 330 miles south of Moscow, seemed designed to bring the war home to Russia, where many do not feel any direct impact of a conflict that has destroyed many Ukrainian cities and towns and displaced millions. It also may be intended to divert Russian troops from other locations along the front, where Ukraine’s military has steadily lost ground in recent months. Analysts suggested Kyiv could be trying to gain leverage for any future negotiations with the Kremlin.

       While previous offensives into Russia were led by volunteer militant groups opposed to President Vladimir Putin but not formally affiliated with Ukraine’s Armed Forces, this attack appears to involve several thousand troops, including some from elite assault brigades equipped with U.S. and German vehicles and tanks. They crossed the border Tuesday from Sumy in northeastern Ukraine, according to Russian officials and pro-war military bloggers.

       The Biden administration said Thursday that the operation is an acceptable use of U.S. weapons to strike inside Russia, conforming with a May policy change that allows Ukrainian commanders to pursue targets there if Russian forces are preparing to launch attacks into Ukraine.

       “As they see attacks coming across the border, they have to be able to have the capabilities to respond,” Pentagon spokeswoman Sabrina Singh told reporters during a news briefing.

       German officials struck a similar tone. A defense ministry spokesman told The Washington Post that Berlin’s declared aim is to “support Ukraine in its defensive struggle against the Russian aggressor,” and pointed to past statements asserting Ukraine’s “right under international law to defend itself against these attacks.”

       An adviser to President Volodymyr Zelensky, who like some other officials spoke on the condition of anonymity to discuss the operation, confirmed the Ukrainian military activity inside Russia and said its forces had seized about 100 square kilometers. The claim could not be independently verified.

       In three days, the Zelensky adviser said, Ukrainian personnel advanced past towns of thousands of people, took hundreds of prisoners, and captured a gas-metering station that Russia uses for energy transactions with Hungary and Slovakia.

       Video posted by a pro-war Russian Telegram channel appeared to show Western military vehicles near Ukraine’s border inside Kursk, describing them as part of the incursion. The footage showed at least one U.S.-provided Stryker and a German Mardar fighting vehicle on the Russian side of the border. Ukrainian forces appeared to have marked their equipment for Kursk with painted white triangles. The Washington Post could not independently verify when the footage was recorded.

       Drone video published on Instagram by a regiment in Ukraine’s military purported to show Russian soldiers in Kursk surrendering. Personnel from Ukrainian units believed to be fighting in Kursk declined to comment, saying operational secrecy was to their advantage.

       Ukraine’s Sumy region has come under repeated bombardment from Russian forces firing across the border. And while Western countries, including the United States, have allowed Ukraine to use donated weapons to preempt or intercept attacks originating from Russia, the assault on Kursk marks the first known instance in which U.S. officials have applied the same justification to a cross-border ground offensive.

       Matthew Miller, a State Department spokesman, said Russia had launched attacks on Ukraine from the Kursk region and that the United States supported Kyiv’s right to defend itself. “In the area where they are operating,” Miller added, “ … we have seen attacks come from there.”

       Kyiv’s forces have moved more than six miles into Russia, Russian military bloggers reported. The acting governor of Kursk, Alexei Smirnov, reported that dozens of civilians have been injured.

       While the precise objective of Ukraine’s incursion is unclear, in addition to potentially diverting Russian troops from Ukraine’s eastern Donetsk region, Kyiv could be trying to gain leverage in future negotiations, analysts said.

       One assessment indicated the Ukrainians are attempting to disrupt Russian forces moving to the Kharkiv region southeast of Sumy, a U.S. official said. Another official said it seems more likely Ukraine wants to overstretch Russian units rather than capture terrain and hold it for the long term.

       “Russia does not have the manpower where they can send reinforcements … while maintaining their current force posture and numbers, both in the Kharkiv area as well as in the east,” the second official said. “You have to pull forces from somewhere.”

       The Kursk offensive has put new pressure on Putin as Russian civilians have had to be evacuated under the same type of bombardment Ukrainians have endured for more than two years.

       Ukrainian officials have asked Washington to let them to use long-range U.S. ATACMS missiles to hit airfields that Russia is using to retaliate against the incursion — a decision that, if approved, could allow Kyiv to hold a portion of Kursk for some time.

       “This will give them the leverage they need for negotiations with Russia — this is what it’s all about,” the Zelensky adviser said.

       On Thursday, gas was still flowing through Sudzha, the last operational shipping point for a pipeline that carries Russian natural gas to Europe via Ukraine. The station was unlikely to be used for leverage, the adviser said, because with the pipeline running through Ukraine anyway, Kyiv could have disrupted flows at any time.

       Ukraine earns lucrative transit fees but has expressed a desire to cut off Russia’s remaining energy business in Europe. Meanwhile, Russia’s National Guard said it had beefed up security around the Kursk nuclear power station, about 40 miles northeast of the town.

       With Kyiv bracing for Russian reprisals, Ukrainian officials on Wednesday ordered about 6,000 people to evacuate from Sumy.

       Russia’s most senior military commander, chief of general staff Valery Gerasimov, informed Putin on Wednesday that the Ukrainian offensive had been halted. But Russian military bloggers painted a drastically different picture, criticizing Russian forces for not better fortifying the border and lamenting the town of Sudzha as lost to the Ukrainian invaders.

       According to Vyortska, an independent Russian outlet that interviewed people from Sudzha and their relatives, many are facing difficulties leaving town.

       “We called the emergency services and they said they are unable to get anyone out,” one woman, identified as Marina told the outlet. “As I understood, they are not able to evacuate and it is very difficult to move around the town as drones are hitting civilians.”

       Mykhailo Podolyak, a Ukrainian presidential adviser, suggested on the country’s national television news broadcast that any military actions on the Russian territory could better Ukraine’s position during future negotiations with Russia to end the war. He did not comment on the Kursk incursion specifically.

       While Ukraine might be trying to divert forces from the east to defend its new offensive in Russia’s Kursk region, Russia has more personnel and maneuverability than Ukraine. Kyiv has been mobilizing thousands of troops in recent months in a ramped-up conscription campaign, but those recruits need to undergo weeks of training before reaching the battlefield.

       Ukrainian officials have expressed some sense of urgency to better their military position before the U.S. presidential elections in November. The last tranche of American security assistance was held up for more than six months by Republicans in Congress — a signal that future military aid is not guaranteed, especially if GOP nominee Donald Trump wins office. Trump has said that he would quickly push the two sides to reach a negotiated settlement of the war.

       Horton, Hudson and Oakford reported from Washington. Serhiy Morgunov and Serhii Korolchuk in Kyiv, Francesca Ebel in London, Kate Brady in Berlin, and Robyn Dixon and Natalia Abbakumova in Riga, Latvia, contributed to this report.