Gathered in a Ukrainian farmhouse, soldiers checked their kits: rifles, machine guns, grenade launchers, spare batteries for radios, red and white flashlights, all that would be needed for a stealthy and daring night assault across the border into Russia.

The soldiers are Russians who have turned against the government of their country’s president, Vladimir V. Putin, and are now fighting for the Ukrainian side by making incursions back into Russia.

Their goal has been to break through a first line of Russian defenses, hoping to open a path for another unit to drive deeper into Russia with tanks and armored personnel carriers.

“We will jump in their trench and hold it,” one of the soldiers, who declined to be identified for security reasons, explained. “Either we take them out, or they take us out.”

By both Ukrainian and Russian accounts, fierce fighting has raged along Russia’s southern border for five days in the most sweeping ground attacks into Russia since its military invaded Ukraine two years ago.

Three Russian exile groups, which are all but openly backed by Ukraine’s military intelligence agency, say the assaults are timed to undermine the sense of stability that underlies Mr. Putin’s quest for a fifth term, in which three days of voting wrap up on Sunday.