There has been a particular outcry among ethnic minorities in remote, economically deprived areas in Siberia, where Russia's professional armed forces have long recruited disproportionately.
Since Wednesday, people have been prepared to queue for hours to cross into Mongolia, Kazakhstan, Finland or Georgia, scared that Russia might close its borders, although the Kremlin has said reports of an exodus are exaggerated.
The governor of Russia's Buryatia region, which is located on the Mongolian border and home to an ethnic Mongol minority, acknowledged on Friday that some had received papers in error and said those who had not served in the army or who had medical exemptions would not be called up.
On Saturday, Tsakhia Elbegdorj, president of Mongolia until 2017 and now head of the World Mongol Federation, promised those fleeing the draft a warm welcome, and bluntly called on Putin to end the war.
"The Buryat Mongols, Tuva Mongols, and Kalmyk Mongols have ... been used as nothing more than cannon fodder," he said in a video message, wearing a ribbon in Ukrainian yellow-and-blue, and referring to three Mongol ethnic groups in Russia.
"Today you are fleeing brutality, cruelty, and likely death. Tomorrow you will start freeing your country from dictatorship."
The mobilisation, and the hasty organisation of so-called referendums on joining Russia in occupied Ukrainian territories this weekend, came hard on the heels of a lightning Ukrainian offensive in the Kharkiv region - Moscow's sharpest reverse of the seven-month-old war.
The anti-war group Vesna called on social media for new demonstrations across Russia on Saturday evening, after more than 1,300 protesters were arrested in 38 towns on Wednesday, according to the independent monitoring group OVD-Info.
The interior ministry of the Russian region of North Ossetia advised people not to try to leave the country for Georgia at the Verkhny Lars frontier post, where it said 2,300 cars were waiting to cross.
