Russian soldiers who surrender to Ukraine will be treated in a "civilised manner", Ukrainian President Volodymyr Zelensky has said.
In his nightly address, Mr Zelensky appealed to Russians to run away or surrender once at the front.
It comes after Vladimir Putin signed a law doubling the punishment for Russian soldiers who desert or disobey orders.
Fresh protests have also broken out over Russia's partial mobilisation of 300,000 extra troops.
Russian human rights organisation OVD-Info reported that 700 people were arrested on Saturday, while more than 1,000 were detained earlier in the week. Unsanctioned rallies are banned under Russian law.
Speaking in Russian - his first language - Mr Zelensky urged Russians to surrender to Ukraine rather than risk being tried as a war criminal after the conflict.
Ukraine will treat deserters in line with international conventions and will not return anyone to Russia if they are afraid of repercussions, he said.
"It is better to surrender to Ukrainian captivity than to be killed by the strikes of our weapons," he added.
President Putin signed a law on Saturday which means Russians who desert, refuse to fight, disobey orders or surrender now face 10 years in prison.
The Russian president's moves are widely seen as an attempt to regain the initiative after his forces suffered setbacks on the battlefield.
Self-styled referendums on joining Russia are continuing in four regions of Ukraine: Luhansk, Donetsk, Kherson and Zaporizhzhia.
Ukraine and the West have condemned the votes as undemocratic, and there have been numerous reports of locals being intimidated into voting by armed Russian soldiers.
The votes would pave the way for Russia to annex the four regions.
Mr Putin's top diplomat, Sergei Lavrov, said that any regions voting to join Russia would receive the country's full protection, and be bound by all laws and doctrines of Russia.
While annexation would not be recognised internationally, it could lead to Russia claiming that its territory is under attack from Western weapons supplied to Ukraine, escalating the war further.
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The partial mobilisation sparked a rush to the border, with some Russians attempting to evade a call-up.
Estonia, Latvia and Lithuania have closed their borders to most Russians, saying they are not prepared to automatically grant asylum to those fleeing the draft.
Estonian Interior Minister Lauri Laanemets was quoted by the Baltic News Service as saying that the invasion of Ukraine was the "collective responsibility of Russian citizens" and that refusing entry to Russians would hopefully "increase discontent" in Russia.