Some Russian men rushed for the exits on Thursday after President Vladimir Putin ordered a partial mobilisation, with traffic at border crossings with Finland and Georgia surging and prices for air tickets from Moscow rocketing.

Putin on Wednesday ordered Russia's first mobilisation since World War Two and backed a plan to annex swathes of Ukraine, warning the West he was not bluffing when he said he'd be ready to use nuclear weapons to defend Russia.

Prices for air tickets out of Moscow soared above $5,000 for one-way tickets to the nearest foreign locations, with most air tickets sold out completely for coming days.

Social media groups popped up with advice on how to get out of Russia while one news site in Russian gave a list of "where to run away right now from Russia." There were long tailbacks at border crossings with Georgia.

"War is horrible," Sergei, a Russian man who declined to give his surname, told Reuters as he arrived in Belgrade, the Serbian capital. "It's okay to be afraid of war and of death and such things."

Sergei said his friends and colleagues were anxious about the possibility of being sent to fight in Ukraine. "They're afraid also," he said. "It's a difficult situation."

A tourism industry source told Reuters that there was desperation as people sought to find air tickets out of Russia.

"This is panic demand from people who are afraid they won't be able to leave the country later - people are buying tickets not caring where they fly to," the source said.

Traffic arriving at Finland's eastern border with Russia "intensified" overnight, the Finnish Border Guard said.

"The number clearly has picked up," the Finnish border guard's head of international affairs, Matti Pitkaniitty, told Reuters, adding that the situation was under control and border guards were ready at nine checkpoints.

Russian police detained more than 1,300 people in Russia on Wednesday at protests denouncing mobilisation, a rights group said.

In the far eastern region of Yakutia, a military commissar ordered a call for mobilisation. (Writing by Caleb Davis and Guy Faulconbridge, editing by Mark Heinrich)