The Baltic trio of Latvia, Lithuania and Estonia have already closed their borders to most Russians following Moscow's war in Ukraine.
Now, as Russians try to leave their homeland after Vladimir Putin's reservist call-up, at least two of them are extending it to those seeking asylum.
Meanwhile, a spokesman for the European Commission -- asked about the rest of the EU -- said it was down to individual countries to decide whether to grant Russians refugee status. He added discussions were under way with EU nations to come to a common position.
The Kremlin on Thursday said reports of an exodus of draft-age men from Russia following Putin's order were "exaggerated".
Estonia's interior minister Lauri Laanemets, speaking on Wednesday, called Russia's invasion of Ukraine “the collective responsibility of Russian citizens” and argued that allowing in those fleeing possible army service would violate European Union sanctions aimed at Moscow.
“Putin’s latest move in some ways reinforces the sanctions we have imposed to date, because it will hopefully increase discontent among the population," Laanemets was quoted by BNS as saying.
"It is no longer just professional soldiers, people from remote regions or convicts who are being sent to the front, but the desire is for everyone to be relegated to cannon fodder,” he said.
The country's foreign minister Urmas Reinsalu said in an email to Reuters: "A refusal to fulfil one's civic duty in Russia or a desire to do so does not constitute sufficient grounds for being granted asylum in another country"
Latvia's foreign minister Edgars Rinkevics wrote on Twitter that their refusal was linked to security concerns. "Due to security reasons, Latvia will not issue humanitarian or other types of visas to those Russian citizens who avoid mobilisation," Rinkevics wrote.
Lithuania's interior ministry said each asylum case would be weighted separately, but added that the country "does not have the purpose and capacity to issue visas on humanitarian grounds to all Russian citizens who apply for them".
Lithuanian Defense Minister Arvydas Anusauskas said Thursday that “being drafted into the army is not enough” of a reason for Russians to get asylum in his country, which borders the Russian exclave of Kaliningrad.
"We as a European Union, in principle, stand in solidarity with the Russian citizens who have the courage and bravery to show their opposition to what the regime is doing, especially when it comes to this illegal war in Ukraine, and especially in the circumstances like the war, like censorship and other restrictive measures that have been imposed in terms of limiting and curtailing the freedom of expression, freedom of media, freedom of expressing different opinion than the official propaganda is pushing on people," said the EU Commission's spokesman Peter Stano.
"And we also feel sympathy with those Russian families who are fearing for their sons, brothers or fathers who are being sent to die in a senseless war, illegal war. So we feel with these people and this is something that the European Union ministers also are taking into account and discussing."
"The bottom line is we are aware of the situation. This is happening already. Half a million of Russians have left the country since the start of this invasion. The brightest people, the most talented people are leaving, but they found their place either in Europe or worldwide. And we are dealing with this problem and we are discussing the ways how to exactly the best in order to show our solidarity and also to ensure the security and safety of the member states."
OK Russians, a non-profit organisation, estimates that more than 300,000 people have left Russia since the war began in February.
A third of those fleeing Russia, according to surveys by the organisation, went to Turkey, Georgia and Armenia. Most of them are young people between 25 and 34 years old, while only 14% of those who left were aged above 45. Some 68% of respondents told OK Russians that they had left "forever or for a long time". Meanwhile, 90% of those who had left, according to the organisation, were involved in politics, and 55% said they had felt some political pressure in Russia before leaving.