More than 3,000 Russians have crossed one checkpoint into Mongolia since Putin announced the call-up of reservists last Wednesday.
Protests, drafting errors and a mass exodus of eligible reservist troops have wreaked havoc with Russia’s recent military mobilisation.
Last week, President Vladimir Putin announced more people were needed to fight Russia's war in Ukraine.
It sparked a surge in online searches for flights and prices soared on routes from Moscow to Istanbul and Belgrade.
At Russia's border with Mongolia, there were lines of vehicles and people. The head of a checkpoint in the town of Altanbulag said more than 3,000 Russians had entered Mongolia via the crossing since Wednesday, most of them men.
Meanwhile, in Dagestan, one of Russia’s poorer regions, anti-war demonstrators clashed with police.
Officers fired warning shots to try to disperse more than 100 people who blocked a highway while protesting Putin's call-up.
The rally, mostly made up of women, chanted "No to war!" in the region’s capital Makhachkala.
It wasn't immediately clear whether someone was detained or injured or whether the efforts to disperse the crowds succeeded.
Some of them argued with the police saying it was Russia that attacked Ukraine, and more than 100 people were eventually arrested.
Dagestan is among several Russian regions identified by independent media as the homeland for hundreds of Russian soldiers killed in Ukraine.
'Catastrophic in Crimea'
Ukrainian President Volodymyr Zelenskyy again called on Russians to not submit to "criminal mobilisation" on Sunday, amid protests against Moscow's call to arms.
"The more citizens of the Russian Federation at least try to protect their own lives, the sooner this criminal war of Russia against the people of Ukraine will end," Zelenskyy said in his evening address.
He said that the situation is "catastrophic" in Crimea where Russia was using mobilisation to conduct a "genocidal policy" as army calls are mostly addressed to the native Crimean Tatars.
"We see that people, in particular, in Dagestan, have begun to fight for their lives.
“We see that they are beginning to understand that this is a question of their lives.
“Why should their husbands, brothers, and sons die in this war? For a war that one man wants. For a war against our people, on our land. He does not send his children to war," Zelenskyy added.
Earlier on Sunday, Zelenskyy portrayed Russian mobilisation -- its first such call-up since World War II -- as a signal of weakness, not strength.
Meanwhile, Ukraine's air force has posted footage of what it says was the 'destruction' of a Russian drone sent to the city of Mykolaiv.
The US has warned Russia's leader that there will be "catastrophic consequences" if Moscow uses any nuclear weapons.