A border guard on Finland’s southeastern border said the number of Russians entering the country was double that a week ago.
The Kremlin claimed reports of men fleeing were exaggerated. However, on the border with Georgia there were queues of vehicles stretching for miles.
One man told the BBC he had waited 12 hours to cross. Georgia is one of the few countries where Russians do not need a visa to enter.
Other destinations reachable by air — such as Istanbul, Belgrade or Dubai — saw ticket prices soar immediately after the military call-up of 300,000 military personnel was announced earlier this week.
The Kremlin said only people who had done their military service and had special skills and combat experience would be called up.
But amid fears of a wider mobilisation, able-bodied men have been looking for loopholes. Dodging the draft in Russia is punishable by up to 15 years in prison.
Some men are attempting to marry women with multiple children in a bid to avoid being sent to the frontline, according to Sibreal Media, a Russian-language news website based in the Czech Republic.
Others are rushing to register themselves as carers for elderly relatives, according to the website.
SibReal Media reported that marriage register offices were “packed” in the republic of Buryatia in Siberia.
A woman said: “Me and my man first went to Ulan-Ude (the region’s capital), but the queue there was far too long.
“So we turned the car and rushed to Ivolginsk. But there are crowds of people here, too.
“Everyone is getting married, or applying to establish fatherhood, and children are getting registered to their fathers.”
Men responsible for multiple children can be exempted from the draft, but they are expected to be formally married.
Anti-war protests on the streets of Moscow and St Petersburg have seen more than 1,000 arrests.
The independent OVD-Info protest monitoring group said that according to information it had collated from 38 Russian cities, more than 1,311 people had been held by late last night.
Unsanctioned rallies are illegal under Russia’s anti-protest laws.
Meanwhile, Russian-backed officials in four occupied regions of Ukraine were today holding “referendums” on joining Russia.
The votes are taking place over five days but have been denounced as illegitimate and a sham by Ukraine and the West.
The European Union has said it won’t recognise the results and has indicated it is preparing a new package of sanctions against Russia.
The votes were called by pro-Russian officials in the self-declared republics of Donetsk and Luhansk, and in Russian-held parts of Kherson and Zaporizhzhia in the south. Together the four regions make up around 18 per cent of Ukraine’s territory.