The U.S. would "welcome" any Russian citizens seeking asylum in this country after fleeing Russian President Vladimir Putin's military draft, White House press secretary Karine Jean-Pierre said Tuesday.

Jean-Pierre made the comments during a White House Press briefing, even as tens of thousands of Russian citizens have surged into neighboring Eastern European countries. A reporter pressed Jean-Pierre about reports of military-age men fleeing across borders or even injuring themselves to avoid conscription.

"I know the White House has drawn a distinction between the Russian government and the people of Russia. Does the president have a message for some of these men who are desperately trying to flee the country?" a reporter asked.

"We are seeing protests in the streets of Russia, we are seeing people sign petitions, and I think the message they are sending to us very clearly is that this war that Putin started … is unpopular," Jean-Pierre said. "There are people out there in Russia who do not want to fight Putin's war or die for it."

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Russian President Vladimir Putin, left, vowed to strengthen Russia's military cooperation with its allies at the International Military and Technical Forum in Patriot Park outside Moscow, Russia, on Aug. 15, 2022. (Sputnik, Kremlin Pool Photo via AP)

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Ukrainian soldiers fire a U.S. Javelin missile during military exercises in the Donetsk region on Jan. 12, 2022. (Ukrainian Defense Ministry Press Service via AP)

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"We believe that, regardless of nationality, they may apply for asylum in the United States and have their claim adjudicated on a case-by-case basis," she added. "We welcome any folks who are seeking asylum, and they should do that."

Putin ordered a partial mobilization of the Russian military last week, drafting up 300,000 reserve troops. The move has led to widespread fear of general conscription amid the country's invasion of Ukraine.

Mongolia, Georgia and Kazakhstan have borne the brunt of the wave of Russian military-aged men leaving the country. Georgia officials stated that daily border crossings have nearly doubled in less than a week.

Ukraine soldiers in Izyum

Ukrainian soldiers sit on infantry fighting vehicles as they drive near Izyum in eastern Ukraine on Sept. 16, 2022. (Juan Barreto/AFP via Getty Images)

"About four to five days ago, there were five to six thousand visitors [from Russia] daily, and now it has increased to about ten thousand," Georgia Interior Minister Vakhtang Gomelauri told reporters Tuesday.

Others fleeing to Mongolia had to wait in lines more than a dozen hours long to be processed. One man, Aleksey, told Reuters that he was leaving behind his wife and children while the draft is ongoing.

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"We are not afraid, but why do we have to fight in Ukraine, why?" he told Reuters. "If other countries would attack Russia, we would fight for our country. But why are we going to Ukraine? For what?"