Ukraine has begun to "push the war out into the aggressor's territory" as the military continues its surprise incursion into Russia, Volodymyr Zelensky has said.

The Ukrainian President confirmed that his country's "warriors" had crossed into Russia in his nightly address on Saturday. “Ukraine is proving that it can indeed restore justice and ensure the necessary pressure on the aggressor," he said.

It is the first time he has acknowledged the six-day offensive, with Ukraine's plans shrouded in secrecy. Russia's Foreign Ministry vowed a “tough response” to the advance into its Kursk border region over the past week, which appeared to catch the Kremlin off guard.

Videos circulating on social media appeared to show troops tearing down Russian flags and raising a Ukrainian one within Russia's borders.

Civilians living within Russia's border areas were being evacuated, with Russian state television showing footage of a tent camp in the city of Kursk. More than 20 temporary accommodation centers have reportedly been set up in the region.

Kursk's regional governor said a Ukrainian missile shot down by Russian air defences fell on a residential building, injuring 13 people. Aleksei Smirnov said that a “Ukrainian sabotage and reconnaissance group” entered the Belovsky district the previous day, but that the situation had been “stabilized”.

The Russian Defense Ministry said that 35 drones were shot down overnight over the Kursk, Voronezh, Belgorod, Bryansk and Oryol regions near the border. It comes after Ukraine increased the pace of similar drone attacks targeting military infrastructure and oil depots in recent weeks.

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Overnight into Sunday, a Russian drone and missile barrage on Kyiv killed a man and his 4-year-old son. The bodies of the 35-year old man and his young boy were found under rubble after fragments of missiles fell on a residential area in Kyiv's suburban Brovary district, according to Ukraine's State Emergency Service.

Another three people in the area were injured in the attack.

Meanwhile, Belarus said that it was sending more troops to its border with Ukraine on Saturday after claiming Ukrainian drones entered its air space. Authoritarian leader Alexander Lukashenko said Belarusian air defense forces destroyed dozens of targets flying from Ukraine over the Mogilev region, which borders Russia, on Friday evening.

Belarusian Defense Minister Viktor Khrenin branded it a provocation and said Belarus was “ready for retaliatory action."