Three Ukraine-based Russian paramilitary groups said that they have crossed into Russia and are now fighting government troops there, media reported.
The Freedom of Russia Legion (FRL) and Siberian Battalion (SB) posted videos purportedly showing their fighters in Russia's Belgorod and Kursk regions, BBC reported.
The FRL and an exiled Russian politician claimed two villages were now in control of "liberation forces".
Russia's Defence Ministry said the breakthrough attempts were thwarted.
It claimed more than 234 Ukrainian troops had been killed and several tanks had been destroyed.
According to Belgorod's regional governor, Vyacheslav Gladkov, one member of Russia's territorial defence forces was killed and 10 civilians were injured, BBC reported.
These numbers have not yet been verified by the BBC.
Ukraine's military denied any involvement in Tuesday's cross-border raids.
Andriy Yusov, a spokesman for the country's military intelligence, said the paramilitary groups were "independent organisations" of Russian nationals, and therefore operating "at home".
In a separate development, Russia said Ukraine had launched 25 drones on targets across Russia, but the attack was thwarted.
However, videos have emerged which appear to show several Russian oil facilities on fire, BBC reported.
In the Ivanovo region, just east of Moscow, an Il-76 military transport plane with eight crew and seven passengers crashed shortly after take-off, Russia's Defence Ministry was quoted as saying by Russian state-run news agencies.
The Ministry said an engine fire caused the crash. It gave no details of survivors.
Videos have emerged purportedly showing the plane on fire circling in the sky, and later plumes of black smoke from the crash site.
Elsewhere, at least three people were killed and 38 injured on Tuesday night when a Russian missile struck two apartment buildings in the central Ukrainian city of Kryvyi Rih, according to the Ukrainian Minister of Internal Affairs, Ihor Klymenko.
Klymenko said children were among the victims and that the number of deaths may go up as the search and rescue operation is continuing, BBC reported.
Ukrainian President Volodymyr Zelensky said the country would "inflict losses on the Russian state in response -- quite rightly".
A full-scale invasion of Ukraine launched by Russian President Vladimir Putin is now in its third year, with no signs that the biggest war in Europe since World War Two could end soon.
On Tuesday, the FRL posted what it said was a video from the Russian-Ukrainian border.
"Like all our fellow citizens, in the Legion we dream of a Russia freed from Putin's dictatorship. But we don't just dream: we make every effort to make these dreams come true. We will take our land away from the regime, centimetre by centimetre, " an armed FRL soldier in the footage is heard saying.
Meanwhile, the SB said "fierce fighting is going on the Russian Federation territory", publishing a clip purportedly showing its fighters engaging with Russian government forces.
It also condemned Russian presidential elections on March 15-17, in which Putin is widely expected to be declared the winner.
"Ballots and polling stations in this case are fiction. You can really change your life for the better only with weapons in your hands, " the SB said.
Another Ukraine-based Russian group, the Russian Volunteer Corps (RDK), also published footage of what it said were its fighters engaging with Russian government troops, BBC reported.
"The army of the Kremlin regime lays down its weapons without even starting the battle, " it said.
The videos have not been independently verified.
The border village of Tetkino in the Kursk region appeared to be one of the targets in Tuesday's raids, with the FRL claiming that "liberating forces" now had full control over the settlement.
The BBC has verified the authenticity of FRL footage depicting a strike on an armoured personnel carrier in Tetkino.
Kursk Mayor Igor Kutsak ordered all schools in the regional capital to be put on remote learning from March 13-15 "in connection with recent events".
He also warned that the "missile alert" regime was still in place in the city, which has a population of more than 400, 000.
Earlier on Tuesday, Ukraine-based Russian opposition politician Illya Ponomarev claimed that the border village of Lozovaya Rudka, Belgorod region, was "under full control of liberating forces".
In a statement later on Tuesday, the Russian Defence Ministry said its forces together with border guards and FSB security service units "thwarted an attempt by the Kiev regime to make a breakthrough" into Russia, BBC reported.
It said enemy fighters -- backed by tanks and armoured personnel carriers -- had tried to invade Russia "simultaneously in three directions in the areas of the settlements of Odnorobovka, Nekhoteevka and Spodaryushino, Belgorod region".
It added that another four attacks had targeted Tetkino, but were "repulsed".
The Ukraine-based Russian armed groups have made several cross-border raids since the full-invasion of Ukraine began.
Last May, Russia's military said a similar attack in the Belgorod region was rebuffed and armed insurgents defeated.