Russia pushed ahead on Thursday with its biggest conscription drive since World War Two, prompting some men to rush abroad, while Ukraine demanded "just punishment" for a seven-month-old invasion that has shaken the world.

President Vladimir Putin's order to mobilise another 300,000 Russians escalates a war that has already killed thousands, displaced millions, pulverised cities, damaged the global economy and revived Cold War confrontation.

The mass conscription may be the riskiest domestic move of Putin's two decades in power after Kremlin promises that it would not happen and a string of battlefield failures in Ukraine.

Anti-war protests in 38 Russian cities saw more than 1,300 people arrested on Wednesday, a monitoring group said. Some had been served summons to report to enlistment offices on Thursday, the first full day of conscription, independent news outlets said.

Prices for air tickets out of Moscow soared above $5,000 for one-way flights to the nearest foreign locations, with most sold out for coming days. Traffic also surged at border crossings with Finland and Georgia.

"Every normal person is [concerned]," said one man, identifying himself only as Sergey, disembarking in Belgrade after a flight from Moscow. "It is okay to be afraid of the war."

A Russian man arriving at Istanbul Airport said he left partly over the Kremlin's decision. "A very poor step it seems to be, and it can lead to lots of problems for lots of Russians," said Alex, grabbing his suitcase at a baggage carousel.

The Kremlin said reports of a mass exodus were exaggerated.

Ukrainian President Volodymyr Zelenskiy urged the United Nations to create a special tribunal and strip Moscow of its UN Security Council veto power as a diplomatic showdown loomed on Thursday in New York.

"A crime has been committed against Ukraine, and we demand just punishment," Zelenskiy, dressed in his trademark green military T-shirt, told world leaders by video at the annual UN General Assembly on Wednesday.

The Security Council has been unable to take significant action on Ukraine because Russia is a permanent veto-wielding member, along with the United States, France, Britain and China.

Russian Foreign Minister Sergei Lavrov will face Ukrainian and Western counterparts when UN Secretary-General Antonio Guterres and International Criminal Court prosecutor Karim Khan brief the 15-member council later on Thursday.

Blasts ahead of referendums

On the ground, Russia's military fired nine missiles on the city of Zaporizhzhia, hitting a hotel and a power station, said regional governor Oleksandr Starukh. At least one person died with others trapped under rubble, he said. Zaporizhzhia is about 50 kilometres from the nuclear plant of the same name.

In the Russian-held southern city of Melitopol, also in the Zaporizhzhia region, a blast hit a crowded market. The city's exiled mayor said it had killed three soldiers and was staged by occupying forces to accuse Ukraine of terrorism, while a member of the Russian-installed local administration accused Ukrainian special services of trying to sow terror on the eve of a vote.

Pro-Moscow regional leaders announced referendums on joining Russia from Friday until September 27 in Luhansk, Donetsk, Kherson and Zaporizhzhia provinces — around 15 per cent of Ukrainian territory.

Ukraine and its allies have denounced the plans as a "sham".

Putin says Russia is carrying out a "special military operation" to demilitarise Ukraine, rid it of dangerous nationalists and defend Moscow from the transatlantic alliance Nato. Kyiv and the West call Russia's actions an unprovoked, imperialist land grab to reconquer a country that shook off Russian domination with the 1991 break-up of the Soviet Union.

Former Russian president Dmitry Medvedev, now deputy chairman of the national security council, reiterated on Thursday Moscow's threat to use nuclear weapons if necessary to protect itself and any incorporated territories.

Moscow does not fully control any of the four regions it is apparently seeking to annex, with only around 60pc of Donetsk and 66pc of Zaporizhzhia regions held by its forces.

Freed by 'skin of teeth'

Ukraine extended its hold on recaptured northeastern territory earlier this week as troops marched farther into areas abandoned by Russia, paving the way for a potential assault on occupation forces in the Donbas industrial heartland.

Russia and Ukraine carried out an unexpected prisoner swap on Wednesday, the largest since the war began and involving almost 300 people, including 10 foreigners and the commanders who led a prolonged Ukrainian defence of Mariupol earlier this year.

"We're now out of the danger zone and on our way home to our families," one of those whom Russian forces released, Britain's Aiden Aislin, said in a video from a plane posted on social media. "By the skin of our teeth," added also-released compatriot Shaun Pinner beside him.

Both men had been sentenced to death by a court in the Russian-backed breakaway region of Donetsk.

In Uzbekistan, where many move or travel to Russia for work, authorities warned of prosecution for citizens joining foreign armies. Russia offered fast-track citizenship to those who sign up and Ukraine said it had captured Uzbeks fighting for Putin.