Wagner mercenaries returned to their base after President Vladimir Putin accepted an amnesty deal which meant its leader Yevgeny Prigozhin would go into exile.

The fighters were given guarantees for their safety and the leader, Yevgeny Prigozhin, will move to Belarus, according to the agreement mediated by Belarusian President Alexander Lukashenko.

Prigozhin, 62, a former Putin ally whose forces have fought the bloodiest battles of the 16-month war in Ukraine, said his decision to advance on Moscow was intended to remove corrupt and incompetent Russian commanders he blames for botching the war.

He was seen leaving the district military headquarters in Rostov-on-Don - hundreds of miles south of Moscow - late on Saturday in a sport utility vehicle. His whereabouts on Sunday was not immediately clear.

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Videos shared on social media from Rostov overnight purportedly showed the mercenaries withdrawing from the city in a convoy of armored vehicles, tanks and coaches to the sound of cheers, chants of "Wagner" and celebratory gunfire from local residents.

"Take care of yourselves," shouted one woman.

Moscow was calm on Sunday, with the Red Square closed but otherwise little evidence of increased security in the streets. Monday has been declared a non-working day to allow time for things to settle.

The capital had told residents to stay indoors and deployed soldiers in preparation for the arrival of the Wagner fighters, who appeared to meet little pushback from the regular armed forces.

Chechen special forces who deployed to the Rostov region to resist the mercenaries' advance were also withdrawing back to where they had been fighting in Ukraine, commander Apty Alaudinov said in a video published on Telegram.

After capturing Rostov - the main rear logistical hub for Russia's attacks on Ukraine - Wagner soldiers had raced hundreds of miles north in what Prigozhin called a "march for justice", transporting tanks and armored trucks and smashing through barricades set up to stop them, before the deal to withdraw was reached.

Under the deal, brokered late on Saturday, Kremlin spokesman Dmitry Peskov said a criminal case opened against Prigozhin for armed mutiny would be dropped, Prigozhin would move to Belarus, and Wagner fighters who rallied to his cause would face no action, in recognition of their previous service to Russia.

Peskov said Lukashenko had offered to mediate, with Putin's approval, because he had known Prigozhin personally for around 20 years.

In a televised address during Saturday's drama, Putin said the rebellion put Russia's very existence under threat.

"We are fighting for the lives and security of our people, for our sovereignty and independence, for the right to remain Russia, a state with a thousand-year history," Putin said, vowing punishment for those behind "an armed insurrection" and drawing parallels with the chaos of 1917 that had led to the Bolshevik revolution.

Peskov declined to say whether any concessions were made to Prigozhin, other than guarantees of safety for him - something he said Putin gave his word to vouch for - and for Prigozhin's men, to persuade him to withdraw all his forces.

Prigozhin railed for months against the military's top brass, especially Defense Minister Sergei Shoigu and the chief of the general staff, Valery Gerasimov, accusing them of incompetence and of withholding ammunition from his fighters as they battled to take Bakhmut in Ukraine.

Led by Prigozhin, a former convict whose forces include thousands of ex-prisoners recruited from Russian jails, Wagner has grown into a sprawling international business with mining interests and fighters in Africa and the Middle East.

This month, he defied orders to sign a contract placing his troops under the command of the Defense Ministry. He launched the rebellion on Friday after alleging that the military had killed many of his fighters in an airstrike. The Defense Ministry denied this.

The revolt came just weeks into the start of Ukraine's strongest counteroffensive drive since Moscow's invasion in February last year.

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