The alert came as a Russian plane-launched missile struck a Ukrainian shopping centre, killing at least nine people and injuring 26 others, authorities said.
The mall in Kostiantynivka, in the eastern Donetsk region, is in the town's residential area.
Thick black smoke rose above it after the strike.
Donetsk regional head Vadym Filashkin said in a Telegram post: "This is another targeted attack on a crowded place, another act of terror by the Russians."
Shelling by Ukrainian forces devastated parts of the town of Sudzha in the Kursk region. (AP PHOTO)
In Kursk, acting governor Alexei Smirnov said on Telegram: "The operational situation in the Kursk region remains difficult."
Social services and civic associations were providing assistance to people forced to flee their homes by the fighting, with the last Russian figure for evacuations in Kursk was 3000.
Little reliable information about the surprise Ukrainian operation has emerged and its strategic aims are unclear.
Ukrainian officials have refused to comment specifically about the incursion, which is taking place about 500km southwest of Moscow.
But a top adviser to President Volodymyr Zelenskyy said on Thursday that border region attacks would cause Russia to "start to realise that the war is slowly creeping inside of Russian territory".
Myhailo Podolyak also suggested the operation would improve Kyiv's hand in the event of negotiations with Moscow.
Russian President Vladimir Putin has cast the Ukrainian offensive as a "major provocation".
The assault came as the Ukrainian army toils to hold at bay an intense Russian push at places on the front line in eastern Ukraine, especially in the Donetsk region.
Putin has made clear he wants to capture the parts of Donetsk that the Kremlin's forces do not already occupy.
The Ukrainian offensive is a "major provocation", President Vladimir Putin says. (AP PHOTO)
Russia declares federal level emergencies when there are more than 500 victims or damage exceeds 500 million rubles ($A8.6 million).
The Institute for the Study of War, a Washington-based think tank, says Ukrainian forces have pressed on with their "rapid advances" deeper into the Kursk region, reportedly going up to 35km beyond the border.
"The lack of a coherent Russian response to the Ukrainian incursion into Kursk ... and the reported rate of Ukrainian advance indicates that Ukrainian forces were able to achieve operational surprise," the Washington-based ISW said late on Thursday.
While Russia's emergencies ministry made the federal-level alert, the defence ministry on Friday said only that the military "continues to repel the attempted invasion" and was responding with air strikes, artillery and troops on the ground.
Ukraine has also kept up its strategy of hitting rear areas with long-range drones, targeting military sites, oil refineries and other infrastructure.
Ukrainian drones attacked Russia's Lipetsk region, which is about 300km from the Ukraine border, on Thursday night, authorities said.
The regional emergencies ministry in Lipetsk said there was a fire at a military airfield where fighter planes including MiG-29s and Su-34s reportedly are based.
Lipetsk governor Igor Artamonov said unspecified electrical infrastructure was damaged and nine people were wounded in the attack.
The Russian defence ministry said 75 Ukrainian drones were shot down during the night, 19 of them over Lipetsk.
