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.

Ukraine has grown increasingly bold recently in staging direct attacks inside Russia, sabotaging railroads in Siberia, hitting refineries and fuel depots with exploding drones and now backing groups driving tanks across the border. Fearing Russia might escalate its military response, the United States and its Western allies have throughout the war prohibited Ukraine’s military from using donated weaponry in these attacks.

Military analysts say the attacks divert Russian air defenses away from the battlefield, put a dent in Russia’s oil economy where sanctions have fallen short, unnerve Russians and can create leverage in any future negotiations, even as Ukraine suffers setbacks along the front inside the country.

The area around the border where the exile groups are assaulting — a thinly populated area of fields, forest and small villages — had already descended into a chaotic state after nearly two years of Russian cross-border attacks with artillery and small sabotage units that had been slipping into Ukraine.

The response, a politically driven escalation ahead of Russia’s election, is distinct for its scale and the number of soldiers involved, commanders in the Russian exile groups said.

A half-dozen Russian border posts and villages have come under attack and the tanks are the first foreign military assaults inside Russia since World War II, the Russian exiles say.

The ground fighting coincided with a flurry of long-range Ukrainian drone strikes on Russian oil refineries and on the Russian city of Belgorod. Two people died in drone strikes in the city on Saturday, the regional governor said.

Mr. Putin, speaking at a security council meeting Friday, described “attacks on peaceful settlements in the territory of Russia” and said 2,500 soldiers, whom he called mercenaries, directed by the Ukrainian government, along with tanks and armored vehicles, were carrying out assaults along the border. The attacks at five sites were intended to disrupt voting this weekend but all had been repelled, Mr. Putin said, adding, “the enemy will not go unpunished for these strikes.”

The three exile groups — Free Russia Legion, the Russian Volunteer Corps and the Siberian Battalion — have declined to disclose their numbers but confirm the use of tanks in the fighting.

“Putin has twice commented on our special liberation operation, which means we are hitting the target,” said Aleksey Baranovsky, a spokesman for the Free Russia Legion. The attacks, he added, were intended to show resistance to Mr. Putin during an otherwise stage-managed election.

“An election is a time for our voices to be heard,” he said.

The attacks are continuing along a roughly 100-mile stretch of border between the Sumy and Kharkiv regions of Ukraine and the Belgorod and Kursk regions of Russia, according to both Russian and Ukrainian sources.

Russian military bloggers have identified nine incursion sites. Both sides described cross-border helicopter assaults staged from inside Ukraine. The preparation for the operation, witnessed by journalists with The New York Times, involved about 50 soldiers, two tanks and four armored personnel carriers, including two American-designed M-113 armored personnel carriers. Many nations have donated M-113s to Ukraine.

Russia’s Ministry of Defense said in a statement Friday that it had repelled all the attacks and used rockets to strike the invading soldiers who had landed by helicopter, forcing them into a minefield. The statement said Russian forces had destroyed 18 tanks and 23 other armored vehicles.

Farther from the border area, Ukrainian drones on Saturday hit two oil refineries in the Samara region, on the Volga River in central Russia, starting a fire at one, regional officials and Russian media reported. Ukraine has struck a dozen or so refineries since the beginning of the year, and Russian media have reported rising gasoline prices inside Russia.

Ukraine has been recruiting among exiled Russian nationalists and disaffected ethnic minorities. The leader of the Russian Volunteer Corps, Denis Kapustin, openly espouses far-right views and uses White Rex as his military call sign. German officials and the Anti-Defamation League have identified Mr. Kapustin as a neo-Nazi.

In an interview at a base in a Ukrainian village on Wednesday, Mr. Kapustin said the attacks inside Russia timed to the election were larger than the sabotage operations with small units he had conducted.

The group, he said, was now attacking “en masse with tanks, armored vehicles and artillery” and had successfully destabilized the border region ahead of the election. Along with a cross-border attack last spring, he said, his group had been able to derail trains in small operations.

His group’s assaults into Russia, he said, had pierced Mr. Putin’s assumption that Russia would be immune from retaliatory attacks after having invaded Ukraine.

“Obviously, they were shocked,” he said of Russia’s leadership. “They realized, OK, the Pandora’s box is open now. Anything can happen.”

Ukraine’s military, he said, “helps us enormously” with intelligence, logistics and evacuation of wounded, but, he added, it does not send Ukrainian citizens across the border into Russia. The operations’ ultimate goal, he said, is more than making “hit-and-run” attacks; it is to hold territory inside Russia.

The cross-border incursions, he said, had forced Russia to divert military resources that could have gone to the front in southeastern Ukraine. Still, Russian troops have an advantage in numbers, weaponry and ammunition and have been creeping forward in the trench fighting in Ukraine.

The Ukrainian-backed groups have suffered setbacks on the border, too. Russian jets have been bombing near the border inside Ukraine, and Ukrainian authorities on Saturday ordered the evacuation of 22 towns and villages.

After clipping on gear and checking their weapons, the unit of the Russian Volunteer Corps that suited up in a farmhouse in the predawn hours Thursday were ready to launch their attack inside Russia — but the tanks and armored vehicles for the second wave of their assault were nowhere to be seen.

The soldiers sat on the floor, and one fell asleep on top of boxes of tank ammunition. The column of armored vehicles had gotten lost on back roads near the border.

“Send me your coordinates,” a commander shouted over the radio at the armored personnel carrier drivers. The drivers did not know them.

“Is it stupidity or sabotage?” the commander yelled back.

Trucks were sent to look for the armored vehicles. Hours passed before they arrived, and it was already dawn, even though the incursion into Russia was to have begun at night. It would now take place in daylight.

“It’s a war, nothing ever goes according to the plan,” one of the officers said.

Later, one tank broke down before reaching the border, and another was hit with a rocket-propelled grenade in the fighting.

The group returned to its base Thursday evening, without having broken through a line of defenses just across the Russian border.