Shelling by Ukrainian forces has killed five civilians in a Russian town near the border, the region’s governor said on Sunday, and local authorities in Ukraine said that three people were also killed in Ukraine by Russian attacks.

The deaths of civilians in areas near the border provide further evidence that the two warring countries have kept up their cross-border fire nearly three weeks into a Ukrainian offensive into Russia’s Kursk region.

The shelling in the small Russian town of Rakitnoye, which is in the Belgorod region and around 15 miles from the border, left 12 people wounded in addition to the five killed, the regional governor, Vyacheslav Gladkov, said in a post on the Telegram messaging app. “A difficult night for the whole region,” he said.

In northeastern Ukraine, Russian forces pounded the Sumy region with more than 74 artillery, mortar and drone strikes over the past 24 hours, the region’s military administration said on social media in a daily report. Three people were killed and nine others were wounded.

There was no independent confirmation of either of the reports.

Daily cross-border shelling in northeastern Ukraine has been a constant backdrop of the war, taking a deadly toll on villages, according to officials from both sides who rarely acknowledge attacks by their own forces. But the military situation has become more acute since Aug. 6, when Ukrainian troops and armored vehicles stormed into the Kursk region of western Russia, swiftly pushing through Russian defenses and capturing several villages.

That offensive was partly aimed at diverting Russian forces from the eastern front in the Donetsk region, the part of the long front line that has been most active this year, but it has slowed in recent days.

President Volodymyr Zelensky of Ukraine said that one aim of the incursion was to demonstrate that President Vladimir V. Putin of Russia was more interested in retaining territory Moscow has occupied in Ukraine than defending Russian villages.

Russia illegally annexed Ukraine’s Crimean Peninsula in 2014 and since its full-scale invasion of Ukraine began in February 2022, it has occupied and illegally annexed parts of the Zaporizhzhia region in southern Ukraine as well as a substantial part of Ukraine’s east.

Mr. Putin “is still thinking about how to keep the occupied territories and does not think about how to protect his people,” Mr. Zelensky said during a news conference on Saturday with the leaders of Poland and Lithuania. He also said that the incursion aimed to stop Moscow’s own plans to launch an offensive in northern Ukraine and to occupy Sumy region.