President Volodymyr Zelensky said "fierce fighting" was under way and governor Oleg Synegubov said "all areas" of the regional border with Russia were now "under enemy fire almost around the clock".
In Moscow, Russian President Vladimir Putin moved to replace defence minister Sergei Shoigu, in what would be a major shake-up of the military leadership.
Across the border in the Russian city of Belgorod, emergency services said eight people were killed when a residential building was hit by a Ukrainian missile after it was intercepted by air defences.
In Ukraine, local prosecutors said four civilians had been killed in the Kharkiv region in the Russian ground offensive, which was launched on Friday.
The Ukrainian army's top commander said that although the situation was "complicated", his forces were managing to hold back further Russian advances.
But Russia's defence ministry said its forces had "advanced deeply into the enemy defences", a day after claiming the capture of five villages in Kharkiv region.
At an evacuation point near the front line in the Kharkiv region, AFP reporters on Sunday saw groups of people evacuated from around the border town of Vovchansk, most of them elderly and disoriented.
"We weren't going to leave. Home is home," said 72-year-old Lyuda Zelenskaya, hugging a trembling cat named Zhora.
Liuba Konovalova, 70, said she had endured a "really terrifying" night before her evacuation.
Volunteers assisted evacuees to a few wooden benches where they registered and received food before being evacuated towards Kharkiv, the regional capital.
In the last few days, nearly 6,000 residents in and around Vovchansk have been evacuated, said Kharkiv governor Synegubov.
"Defensive battles and fierce fighting continue on a large part of our border," Zelensky said in his evening address.
"There are villages that have actually turned from a 'grey zone' into a combat zone, and the occupier is trying to gain a foothold in some of them, or simply use them for further advancement," he said.
A "grey zone" is territory not fully controlled by either side.
Zelensky added the situation was "extremely difficult" around Vovchansk.
- 'Everything is being destroyed' -
Speaking at an evacuation point near Vovchansk, Oleksiy Kharkivsky, a senior police officer helping to coordinate evacuations, said the area was "constantly under fire".
"Everything in the city is being destroyed... You hear constant explosions, artillery, mortars. The enemy is hitting the city with everything they have," he said.
Officials estimated there were still around 500 civilians left in Vovchansk as Russian troops close in.
Volodymyr Tymoshko, head of police in the Kharkiv region, said the town was being attacked on three sides and Russian troops were on the outskirts.
"Despite active fighting, the police are still taking out people who live some 300, 500 metres from the contact line at the moment," he told AFP at the evacuation point.
Ukraine's commander-in-chief Oleksandr Syrsky said on social media that his army's defences were holding but conceded the situation in the Kharkiv region had "deteriorated significantly".
- 'Return the initiative to Ukraine' -
In Kharkiv itself, mayor Igor Terekhov was quoted by the city council as saying there was no reason for people to leave the city despite the offensive.
"Despite all the events that are taking place in the region, Kharkiv is calm. We do not see people leaving," he was quoted as saying.
On Saturday, AFP saw groups of people fleeing the border area arriving in vans and cars loaded with bags at a reception centre for evacuees near Kharkiv.
Evacuees -- most of them elderly -- received food and medical assistance.
Zelensky said on Saturday that Ukrainian troops had been carrying out counterattacks in the border villages.
"Disrupting Russian offensive plans is now our number one task," he said.
Troops must "return the initiative to Ukraine", the president insisted, again urging Western allies to speed up arms deliveries.
Ukrainian officials had warned for weeks that Moscow might try to attack its northeastern border regions, pressing its advantage as Ukraine struggles with delays in Western aid and manpower shortages.
Putin's announcement that Shoigu was being removed as defence minister is expected to be approved by Russia's parliament on Tuesday.
His replacement, Andrey Belousov, has no military background. He has been one of Putin's most influential economic advisers over the last decade.
Shoigu has been named as secretary of the Security Council, replacing long-standing Putin ally Nikolai Patrushev.
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