Tens of thousands of protesters chanting 'Russia Will be Free' rallied in Moscow on Tuesday against President Vladimir Putin's third term despite a police crackdown on their leaders a day earlier.

Flag-waving crowds made their way down leafy boulevards from Moscow's central Pushkin Square as the march began with police putting the numbers at around 18,000.

City authorities have greatly played down previous protest figures. Opposition leader Sergei Udaltsov put the turnout at over 100,000, although reporters on the ground said the figure was more in the tens of thousands.

Some shouted slogans such as 'Russia without Putin' and held up whimsical protest symbols that have defined the light-hearted mood of historic protests which gripped Moscow this winter.

Almost all the main leaders of the social network driven protest were absent as they were being questioned by investigators, who hauled them in in an apparent bid to disrupt the march.

It was the first mass rally after the ruling party rammed through legislation raising penalties for offences by protesters to 300,000 rubles ($9,000) -- about the size of Russians' average annual pay.

The sanctioned march, to be followed by a rally, came a day after masked officers carried out early morning raids on the protest leaders' apartments as part of a criminal probe into violence at an earlier protest.

They searched the homes of the charismatic lawyer and anti-corruption blogger Alexei Navalny and television presenter Ksenia Sobchak, who played a major part in recent Occupy-style protests.

Others targeted included the moderate democracy campaigner Ilya Yashin, who was staying at Sobchak's apartment, as well as Udaltsov.

Protesters on Tuesday responded to the heavy-handed tactics with posters saying 'No to 1937,' referring to Stalin-era persecutions, and "Stop the repression, you are making us into revolutionaries."

Navalny, Yashin and Sobchak were all released after being questioned at the Investigative Committee on Tuesday morning and were expected to join the rally later in the day.

"Investigators intend to find out their role in organising (the May 6) mass demonstration," said Investigative Committee spokesman Vladimir Markin.

He later added that the protest leaders were questioned as witnesses rather than suspects in the unrest.

The investigation concerns bloody battles that broke out between riot police and the mostly young crowd during a Moscow event held on the eve of Putin's May 7 swearing in, although Sobchak did not even attend.

Scenes of police clubs swinging against the backdrop of the Kremlin drew a sharp rebuke from Western governments and dealt a bruising political blow to Putin on the very first day of his return from a four-year stint as premier.

Udaltsov turned down a summons for questioning and attended the march, saying that he was obliged to as one of the organisers.

"I will attend voluntarily and will answer all the questions," he said at the start of the march, wearing dark glasses and looking drawn after recent detentions.

"They wanted to behead the protest movement" said opposition lawmaker Dmitry Gudkov, who is one of the protest organisers, at the march.

"No one is afraid of arrests since we can't live this way any longer."

"We need to make the law the governing principle, that is what I hope these rallies accomplish," said one of the protesters, art teacher Vera Fyodorova.

The chief of Putin's human rights council, Mikhail Fedotov sounded the alarm on the raids on Monday, saying that "from the standpoint of social harmony, modernisation and political reforms, this is the very worst that could have happened."

US State Department spokeswoman Victoria Nuland told reporters that Washington "is deeply concerned by the apparent harassment of Russian political opposition figures on the eve of the planned demonstrations."