BERLIN — Austria’s troubled coalition government descended deeper into chaos on Monday, with the far-right Freedom Party pulling out its remaining ministers and casting adrift important parts of the government.

The fresh turmoil came after the governing coalition collapsed this weekend, when a video was made public appearing to show the vice chancellor, Heinz-Christian Strache of the Freedom Party, promising government contracts to the niece of a Russian oligarch close to President Vladimir V. Putin. Mr. Strache quit on Saturday.

The scandal has raised concern in Austria and beyond of the influence the Kremlin wields with sympathetic far-right and populist parties around the Continent, just days before elections for the European Parliament in which nationalist forces are predicted to make a strong showing.

In response to the scandal, Chancellor Sebastian Kurz said on Monday that he would ask the president to dismiss the interior minister, Herbert Kickl, a leading member of the Freedom Party. Mr. Kurz said he had failed to grasp the gravity of the scandal and failed to take firm action in its wake.