Ukraine staged a flurry of cross-border ground attacks with tanks and other armored vehicles and long-range drone strikes into Russia on Tuesday, assaults that appeared aimed at disrupting President Vladimir V. Putin’s re-election campaign messaging that the war had turned in Moscow’s favor.

Three armed groups of Russian exiles who operate in coordination with Ukraine’s military said they had crossed the border into southern Russia overnight and were fighting in border regions. Farther from the border, drone strikes hit a Russian oil refinery and fuel depot.

Throughout the war, Ukraine has struck targets inside Russia to disrupt military logistics, hit airplanes parked on runways and blown up railway bridges. The cross-border attacks, Ukrainian officials have said, are also intended to unnerve Russians and undermine Mr. Putin’s efforts to insulate them from the war.

Mr. Putin has through his two and a half decades in power — and through multiple elections, the next of which is scheduled to be held over the weekend — portrayed an image of bringing order to Russia. The Kremlin has also barred the only vocally antiwar candidate from running.

The reported border-area fighting in two regions, Kursk and Belgorod in southern Russia, could not immediately be independently confirmed.

The groups saying they crossed into Russia — the Free Russian Legion, the Russian Volunteer Corps and the Siberian Battalion — coordinate with Ukraine’s military. Some members of the groups, including the leader of the Russian Volunteer Corps, hold far-right nationalist views.