Satellite images have shown huge queues at the Russia-Georgia border
The number of Russians arriving in Georgia has nearly doubled since Vladimir Putin's call-up of extra reservists to fight in Ukraine.
Georgia's interior ministry says 10,000 are arriving each day, compared with up to 6,000 before Moscow's announcement.
"The number has increased to some 10,000 a day," said the ministry. "For example, there were 11,200 on Sunday and fewer than 10,000 on Monday", against "5,000 to 6,000" just before the Russian government decree on 21 September.
It came as Moscow ally Kazakhstan vowed to protect Russians that had arrived in recent days.
"In recent days, lots of people come to us from Russia. Most are forced to leave because of a hopeless situation," said Kazakhstan president Kassym-Jomart Tokayev.
"We must take care of them, ensure their safety."
Russia's borders have seen an outflow of military-aged men since the partial mobilisation was declared last week.
Independent newspaper Novaya Gazeta reported that 261,000 men had left the country since the decree was issued, citing an unnamed source in Russia's presidential administration, according to Reuters.
Over the weekend, 17,000 Russians entered Finland, but Baltic neighbours Estonia, Latvia and Lithuania say they will turn away Russians at the border hoping to avoid mobilisation.
On Monday, a senior Russian lawmaker said the country's borders should be closed to draft-eligible men amid the exodus.
"Everyone who is of conscription age should be banned from travelling abroad in the current situation," Sergei Tsekov, a member of Russia's upper house of parliament, the Federation Council, told RIA news agency.
However, Kremlin spokesman Dmitry Peskov said no decisions had been taken on closing Russia's borders.
"I don't know anything about this. At the moment, no decisions have been taken on this," Peskov said.
On Monday, the Kremlin admitted Russia's military mobilisation decree had been "violated", with some men called up in error.
There have been reports of elderly or medically exempt men being called up for service in Ukraine.
Kremlin spokesman Dmitry Peskov said that some draft orders had been issued in error, saying mistakes were being corrected by regional governors and the ministry of defence.
"There have been cases when the decree is violated ... These cases of non-compliance with the required criteria are being eliminated," Peskov said.
The mobilisation process is already underway in Russia with men being deployed ahead of joining the conflict in Ukraine.
Russia's Defence Ministry said about 300,000 people would be summoned to active duty but there are reports that the final number will be higher.
On Monday, a young man shot a Russian military officer at close range at an enlistment office in Siberia after saying “no one will go to fight” and “we will all go home now", according to AP.
It followed arson attacks at other enlistment offices and widespread protests against mobilisation that have seen hundreds arrested.