Both sides are trying to sap morale and weaken military capacity ahead of a long-promised Ukrainian counter-offensive against Russia's 15-month-old invasion.
The war has killed tens of thousands, shattered Ukrainian cities and brought increasing attacks on Russian soil.
Russia's defence ministry said it had repelled three cross-border attacks on Thursday near the town of Shebekino, and it accused Ukraine of using what it said were "terrorist formations" to carry out attempted attacks on Russian civilians.
"The selfless actions of Russian servicemen repelled three attacks by Ukrainian terrorist formations," the Russian ministry said. "No violations of the state border were allowed."
The Russian Volunteer Corps (RVC), a far-right paramilitary group of ethnic Russians that supports Ukraine, claimed to be fighting inside Russia. "The second phase promised by the RVC's commander has begun!" it said on Telegram, referring to a previous incursion, alongside images of fighters firing weapons and inside a building.
Ukraine denies its military is involved in the incursions into Russia's Belgorod border region and says they are conducted by Russian volunteer fighters.
Moscow accuses Ukrainian "terrorists" of targeting Russia's western border. Russian officials say the group of fighters is a proxy run by Ukraine.
The Belgorod region's governor, Vyacheslav Gladkov, said Ukraine's armed forces had repeatedly shelled Shebekino with Soviet-designed Grad 122mm rockets, setting alight a dormitory and damaging an administrative building.
At least nine civilians were injured, he said, with hundreds of children, women and elderly being evacuated. Unverified video showed a fire at a large building in Shebekino.
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