Georgia and Kazakhstan said on Tuesday that tens of thousands of Russians had flooded into their countries from neighbouring Russia since the announcement of partial mobilisation to fight in Ukraine.

Russian President Vladimir Putin last week announced the call-up of thousands of reservists, sparking protests across the country and a rush among Russian men for the borders.

Fyodor said he had fled to Russia's border with Kazakhstan spooked by reports that even the infirm and elderly were being called up to fight.

"There is full chaos (in Russia)," the 24-year-old said. "We don't understand what will happen."

He decided to leave for Kazakhstan on Saturday morning "as a precautionary measure" to "take a head start, just in case."

On Tuesday Kazakhstan said around 98,000 Russians had entered the country since mobilisation was announced.

It took Fyodor about 48 hours, including a five-kilometre walk to the border and a six-hour queue, before he reached the northern Kazakh city of Oral.

"It was raining, it was cold, but six hours of wait... well, that was still reasonable given the circumstances," he said.

Vladislav, a 25-year-old bartender, found shelter in the Kazakh capital of Astana on Monday evening.

In Russia, he said, "I could go to work or to do the groceries and never come back... I don't want to die."

"A week ago, I could not imagine I'd be in Kazakhstan," he said, adding that he "wanted to thank the Kazakh people for welcoming him so nicely."

Kazakh President Kassym-Jomart Tokayev said on Tuesday his country would ensure the safety of Russians fleeing "a hopeless situation".

"This is a political and humanitarian issue," Tokayev added.

"The territorial integrity of states must be unshakeable," Tokayev said.

Kazakhstan has condemned Moscow's assault on Ukraine and called for respect of territorial integrity, as Russia held annexation referendums in four Ukrainian regions.

"In our immediate vicinity a major war is underway. We must remember this, thinking above all about our security," he added.

Russians also flocked to the Black Sea nation of Georgia.

On Tuesday, Georgia said the number of Russians arriving each day has nearly doubled since the draft was announced.

"Four to five days ago 5,000-6,000 (Russians) were arriving in Georgia daily. The number has grown to some 10,000 per day," Interior Minister Vakhtang Gomelauri told journalists.

Georgia and its neighbour Armenia, which do not require visas for Russians, have been major destinations for Russians fleeing since the war began on February 24.

Over the first four months of the war, nearly 50,000 Russians fled to Georgia and another 40,000 to Armenia.

On Tuesday, the local interior ministry in a Russian region that borders Georgia said there was a tailback of around 5,500 cars waiting to cross the Georgian border, calling the situation "extremely tense".

The ministry added that a mobile draft office will be set up at the border in the "near future."