More than 4,100 Russians crossed the border without authorization in the 2021 fiscal year, nine times more than the previous year. This year, the numbers are even higher — 6,420 during the first four months alone.

Ukrainians have also been crossing in greater numbers, with 1,000 apprehensions in the first four months of fiscal 2022 — some as recent as this week — compared with 676 in 2021.

Like Ms. Shuvalova and Mr. Ignatev, many of the newly arriving Russians are supporters of the jailed Russian opposition leader Aleksei A. Navalny and said they no longer felt safe in their homeland. They include L.G.B.T.Q. people and religious minorities, such as Jehovah’s Witnesses, who were ostracized and harassed.

“I get calls every other day; people have been fleeing Russia like crazy,” said Anaida Zadykyan, an immigration lawyer in Los Angeles who has been helping Russians file asylum claims.