Russia is evacuating residents from a second border region, as Ukraine continues its surprise week-long offensive inside the country.
Some 11,000 people in the Belgorod region have been moved, Russian state media reported, because of "enemy action" near the border.
At the weekend, President Volodymyr Zelensky said Ukraine was taking the war into "the aggressor's territory".
He acknowledged the attack for the first time on Saturday night - saying Russia had launched 2,000 attacks from Kursk this summer and it deserved a response.
A senior Ukrainian official told the AFP news agency that thousands of troops were engaged in the operation, far more than the small incursion initially reported by Russian border guards.
The official added that there aim was to "to inflict maximum losses and to destabilise the situation in Russia".
Video from social media and verified by Reuters showed Ukrainian soldiers raising their flag in the Russian village on Guevo, in Kursk.
On Monday morning, residents in parts of Belgorod - which lies to the south of Kursk - were told not to panic but that they must evacuate.
Belgorod governor Vyacheslav Gladkov said people from Krasnaya Yaruga district were being moved due to "enemy activity on the border".
He also said the whole region was under a missile alert, and told people to shelter in their basements.
But he added he was "sure that our servicemen will do everything to cope with the threat that has arisen".
People were also continuing to be evacuated from Kursk on Monday - with thousands of people told to leave their homes in the Belovsky district.
Governor Alexei Smirnov also issued a missile warning - saying people needed to take shelter in rooms without windows and with solid walls.
Temporary accommodation has been prepared, he said.
Some in Russia have questioned how Ukraine was able to enter the Kursk region - with one pro-Russian war blogger, Yuri Podolyaka calling the situation "alarming".
Foreign Ministry spokeswoman Maria Zakharova said a tough response from Russia's armed forces "will not take long".
Meanwhile, Russian ally Belarus said it was bolstering its own troop numbers at its border after claiming Ukraine had entered its airspace with drones.
Belgorod lies next to Kursk - where Ukrainian troops launched their surprise attack into Russian territory last Tuesday.
Ukrainian forces have since advanced up to 18 miles (30km) inside Russia - the deepest incursion into the country since Moscow's full-scale invasion of Ukraine. Kyiv claims thousands of troops are involved.
It marks the biggest co-ordinated attack on Russian territory by Kyiv's conventional forces.
Russia has so far struggled to halt the Ukrainian advance, with more than 76,000 people evacuated from the Kursk region, where a state of emergency has been declared by local authorities.
Earlier this week, Russian President Vladimir Putin described the offensive as a "major provocation".