Hundreds of people have been arrested by authorities as protests against Russia's new "partial mobilisation" continue across the country, an independent rights group has said.

Widespread demonstrations have broken out since President Vladimir Putin announced plans to draft 300,000 men to fight in Ukraine.

The president also signed orders granting Russian citizenship to any foreign national who signs up to serve a year in the country's military.

Elsewhere, other young Russians continue to flee mobilisation by seeking to leave the country.

Meanwhile, Finland has also seen a sharp increase in the number of Russians seeking to enter the country.

Matti Pitkaniitty, a spokesperson for the country's Border Guard, said said the number of Russians arriving had more than doubled since last week.

"The aspiration and purpose is to significantly reduce the number of people coming to Finland from Russia," President Sauli Niinistö told the state broadcaster.

Several other neighbouring states have already ruled out offering asylum to Russians seeking to avoid the draft.

"Many Russians who now flee Russia because of mobilisation were fine with killing Ukrainians," Latvia's Foreign Minister Edgars Rinkēvičs said.

Hundreds arrested as Russian draft protests continue