Ukraine’s president, Volodymyr Zelenskiy, has said his country’s forces are continuing to advance into Russian territory after their surprise offensive, as Kyiv said it had launched a “major” drone attack on four Russian airbases.

Zelenskiy said Ukraine’s troops had advanced several kilometres in the largest attack on Russia since the second world war. Ukraine also claimed to have shot down a Russian Su-34 jet overnight in the Kursk region, where it said it had also captured 100 Russian prisoners.

The claims of the drone strike targeting airbases appeared to be confirmed by Russia, which said it had downed 117 incoming Ukrainian drones overnight.

The strike targeted Russia’s Voronezh, Kursk, Savasleyka and Borisoglebsk airbases, which warplanes use for glide bomb attacks on Ukraine.

Zelenskiy had previously said Ukraine “controlled” 74 Russian settlements, although it was unclear whether or not that meant they were fully occupied by Ukrainian troops.

“Now all of us in Ukraine should act as unitedly and efficiently as we did in the first weeks and months of this war, when Ukraine took the initiative and began to turn the situation to the benefit of our state,” Zelenskiy said in his nightly address on Tuesday.

“Now we have done the exact same thing – we have proven once again that we, Ukrainians, are capable of achieving our goals in any situation – capable of defending our interests and our independence,” he said.

Ukrainian state television aired footage on Wednesday of its troops pulling down a Russian flag from an official building in the town of Sudzha in the Kursk region. The report showed burnt-out Russian military columns on roads in the area as well as Ukrainian soldiers handing out humanitarian aid to residents.

“The situation still remains difficult,” said Yuri Podolyaka, a Ukrainian-born pro-Russian military blogger. “The enemy still has the initiative, and so, albeit slowly, it is increasing its presence in the Kursk region.”

During the major cross-border incursion, which is entering its second week, Ukraine has attacked several border regions as Russia continues to struggle to respond, despite diverting troops from fighting in Ukraine’s eastern Donbas region to counter the advance.

On Wednesday, under heavy shelling by Ukrainian forces, Russia’s Belgorod border region joined Kursk in declaring a state of emergency. The Belgorod governor, Vyacheslav Gladkov, described the situation as “extremely difficult and tense” as he said attacks had destroyed homes and caused civilian casualties.

On his Telegram channel, he said efforts were being made to evacuate minors, with about 5,000 children being relocated to camps in safe areas.

Ukrainian officials have said Kyiv has no plans to occupy Russian territory and that the objective is to prevent Russian missile fire into Ukraine.

Volodymyr Artyukh, the governor of Ukraine’s Sumy region, said there were signs that attacks on civilians in parts of the border area were lessening because of the incursion. “Today, in the districts where the hostilities are taking place, the artillery weapons or the mortars can’t reach us,” Artyukh said in a press briefing, adding that it might now be possible to repair damaged infrastructure in some border areas. “We couldn’t send repair teams there, because they were also targeted by the saboteur groups by drones, aviation or helicopters before, and now it is possible,” he added.

The governor said that civilian officials had had no warning of the Ukrainian incursion into Russia, saying he had found out the “same time as you”. He has ordered an immediate evacuation of 7,000 people living between 5km and 10km of the affected border areas. Officials have said nearly 4,000 people have been evacuated so far while Artyukh said the risk of long-range Russian attacks remained, via air-launched glide bombs or missiles.

Analysts say Kyiv’s forces have targeted the Kursk region because Russia’s weak command and control structure there made it vulnerable.

“The situation is still highly fluid, but with clear signs that the Russian command and control of responding units is still coming together, with all-important unity of command not yet achieved,” said retired US V Adm Robert Murrett, a professor and deputy director of Syracuse University’s security policy and law institute. “The next two or three days will be critical for both sides.”

The surprise offensive began on 6 August and has rattled the Kremlin, amid estimates it could involve as many as 10,000 Ukrainian troops backed by armour and artillery. Despite Kyiv’s stated aims, the operation also appears to be designed to weaken Russia’s offensive in eastern Ukraine by drawing Moscow’s forces away and disrupting supply lines, removing the pressure on several locations where Russia had been making some progress.

That has posed a challenge to Moscow on whether or not to pull troops from the frontline in Ukraine’s east, where achieving a breakthrough is a primary war goal for the Kremlin, to defend Kursk and stop the incursion from ballooning.

The Washington-based Institute for the Study of War thinktank suggested the incursion was unlikely to shift the dynamics of the war. “Russian authorities will likely remain extremely averse to pulling Russian military units engaged in combat from [Donetsk] and will likely continue deploying limited numbers of irregular forces to Kursk … due to concerns about further slowing the tempo of Russian operations in these higher-priority directions,” it said on Tuesday.

According to Russian military bloggers, several irregular units that had been deployed to fight in Donetsk were being sent to Kursk, including the so-called Russian Volunteer Corps and a drone unit associated with a Russian biker gang.

The US president, Joe Biden, said on Tuesday that the developments were “creating a real dilemma” for Vladimir Putin.

The operation has at the very least caused embarrassment for the Kremlin as more than 100,000 Russian civilians have been evacuated. The White House said Ukraine did not provide advance notice of its incursion and the US had no involvement in the operation, though Russian officials have suggested Ukraine’s western backers must have known of the attack.

A woman in Belgorod told the Associated Press on Tuesday that the Ukrainian shelling had been intense for about 10 days until Monday, when there was a lull. The number of people in the region who openly supported the war had decreased after the Ukrainian attacks, the woman told AP, speaking on condition of anonymity.

“When explosions started near the city, when people were dying and when all this started happening before our eyes … and when it affected people personally, they stopped at least openly supporting [the war],” she said.

On Monday, Putin said Ukraine “with the help of its western masters” was aiming to improve Kyiv’s negotiating position ahead of possible peace talks and to slow the advance of Russian forces.