Citizens in four regions of Ukraine were voting for a fourth day on Monday in the Russian-organised referendums that Kyiv and the West have branded a sham. They say the outcomes are pre-determined and they will not recognise the results.

But by incorporating the four regions - Luhansk and Donetsk in the east and Zaporizhzhia and Kherson in the south - Moscow could portray Ukraine's efforts to retake them as attacks on Russia itself, a warning to Kyiv and its Western allies.

US National Security Adviser Jake Sullivan said the United States would respond to any Russian use of nuclear weapons against Ukraine.

"If Russia crosses this line, there will be catastrophic consequences for Russia," Sullivan told NBC television on Sunday. "The United States will respond decisively."

His comments followed Wednesday's thinly veiled nuclear threat by President Vladimir Putin, who said Russia would use any weapons to defend its territory.

Ukrainian President Volodymyr Zelenskiy said he did not believe Putin was bluffing when the Kremlin leader said Moscow would be ready to use nuclear weapons to defend Russia.

The governor of Luhansk, Serhiy Gaidai, said Russian-backed officials were carrying ballot boxes from door to door, accompanied by security officials.

Residents' names were taken down if they failed to vote correctly or refused to cast a ballot, he said.

"Representatives of the occupation forces are going from apartment to apartment with ballot boxes. This is a secret ballot, right?" Gaidai said in an interview posted online.

The four regions represent about 15 per cent of Ukraine. Russian forces do not control all the territory in those regions, where fighting still rages. They would add to Crimea, annexed by Russia in 2014 after a similar referendum there.

Voting ends on Tuesday and Russia's parliament could then move swiftly to formalise the annexations.

Ukraine, bolstered by sophisticated Western weapons, has recaptured swathes of territory over the past month, prompting Putin last week to order Russia's first military mobilisation since World War II to enlist 300,000 additional troops.

That move has unleashed protests across Russia and sent many men of military age fleeing. Almost 17,000 Russians crossed the border into Finland over the weekend, Finnish authorities said on Monday.

More than 2000 people have been detained across Russia for protests at the draft, says independent monitoring group OVD-Info. With criticism of the conflict banned, the demonstrations were among the first signs of discontent since the war began.

A 25-year-old gunman opened fire at a draft office on Monday in the Irkutsk region of Siberia, the local governor said.

In Russia's Muslim-majority southern region of Dagestan, police clashed with protesters, leading to the detention of at least 100 people.

In a Sunday video clip, Zelenskiy addressed Russian protesters directly: "Keep on fighting so that your children will not be sent to their deaths - all those that can be drafted by this criminal Russian mobilisation."

"Because if you come to take away the lives of our children ... we will not let you get away alive."

Separately, Zelenskiy said Ukraine had discovered two more mass burial sites containing the bodies of hundreds of people in the northeastern town of Izium, part of territory recaptured from Russian forces this month.

The seven-month war in Ukraine has killed tens of thousands, flattened towns and cities, fuelled global inflation and triggered the worst confrontation between Moscow and the West since the 1962 Cuban missile crisis, when the United States and the Soviet Union came closest to nuclear conflict.

Australian Associated Press