The Russian military said on Thursday that at least 10,000 people had volunteered to fight in Ukraine, some 24 hours after President Vladimir Putin ordered a mobilisation of reservists.

"During the first day of partial mobilisation, about 10,000 citizens arrived at recruitment offices of their own accord without waiting for a summons," Vladimir Tsimlyansky, a military spokesman, told Russia's Interfax news agency.

He added the military had established a call centre to answer questions from individuals and organisations about the mobilisation.

Amateur footage posted on social media since Putin made the announcement has purported to show hundreds of Russian citizens across the country responding to military summons.

In the Siberian city of Yakutia, footage showed men with their faces obscured embracing family members – some in tears – before boarding a bus.

Popular Telegram channel Mash distributed footage showing long lines of people waiting to board a plane in the eastern Khabarovsk region as part of the mobilisation drive.

A video purportedly filmed in the southern republic of Chechnya showed dozens of young men at a local recruitment office flanked by police.

Putin on Wednesday announced a "partial mobilisation" of reservists and defence minister Sergei Shoigu said the drive would ultimately see 300,000 people called up for service.

More than 1,300 people were arrested across Russia on Wednesday during improvised demonstrations against the mobilisation.

There have also been reports of an accelerated exodus of military-aged men from the country.

Armenia says at least 40,000 Russians have fled there since the beginning of the war, while statistics in June showed nearly 50,000 had fled to Georgia.

The Kremlin on Thursday dismissed as "fake" reports that Russians eligible for mobilisation were rushing for the exit.

"A great deal of false information has emerged about this," said spokesman Dmitry Peskov.

But flights out of Russia were nearly fully booked for the week ahead to cities in the nearby former Soviet countries of Armenia, Azerbaijan, Georgia, Kazakhstan, Kyrgyzstan and Tajikistan. (AFP)