Ukraine's eastern city of Lysychansk is at the centre of competing claims, with both Russian and Ukrainian forces saying they are in control.
Ukraine says its forces are enduring intense Russian shelling there but insists the city has not been seized.
However, Russian-backed separatists say they have successfully entered the city and reached its centre.
Russian media showed videos of separatist or Russian forces apparently parading through the streets.
Russian sources have also tweeted video of the Soviet flag allegedly being placed on the city's ruined administrative centre, but that has not been verified.
It is the last Ukrainian-held city in Luhansk, part of the industrial Donbas region. Russia captured the nearby city of Severodonetsk last month.
The governor of the Luhansk region, Serhiy Haida, said there had been no let-up in the assault on Lysychansk, with Russian forces approaching the besieged city from all sides.
Rodion Miroshnik, ambassador to Russia of the pro-Moscow breakaway Luhansk People's Republic, told Russian television that Lysychansk had been "brought under control" but was "not yet liberated".
Images of Chechen Russian soldiers inside the city were shared by defence blogger Rob Lee.
Since Russia invaded on 24 February, claiming it wanted to "demilitarise" and "de-Nazify" Ukraine after it moved closer to Nato, thousands of civilians and combatants have been killed or wounded, while at least 12 million people have fled their homes.
Western states have responded by arming Ukraine and placing unprecedented sanctions on Russia, a nuclear superpower and global energy supplier.
In another development, railway tracks and electricity lines in the northern city of Kharkiv were damaged in a series of attacks. No casualties were reported.
The southern city of Mykolaiv - on a key route to the port city of Odesa - was shaken by several explosions.
The Russian defence ministry said its air force had destroyed five Ukrainian command posts and several ammunition dumps, but that claim has not been independently verified.
The blasts came a day after the Russians were accused of killing more than 20 people in a missile strike on a block of flats near Odesa.
Later on Saturday, Belarus President Alexander Lukashenko said his country's air defences had shot down Ukrainian missiles, though he did not specify where. He is a close ally of Russia's Vladimir Putin and allowed Russian troops to invade Ukraine from Belarus in February.
"They are provoking us... Three days ago, maybe a bit more, an attempt to strike military facilities on Belarusian territory was made from the territory of Ukraine," he said. "But, thank God, the Pantsir anti-aircraft systems managed to intercept all the missiles."
He added that "we are not seeking to fight in Ukraine".
In its latest intelligence update, the UK Defence Ministry accuses Russia of using Soviet-era anti-ship missiles "in a secondary land attack role" - not what they were designed for. The Kh-22 and Kh-32 missiles were "likely" the ones that killed many civilians in Kremenchuk and Odesa, the ministry says.
Slovyansk, a major Donbas city held by Ukrainian forces, has also been shelled again by the Russians. Its mayor Vadym Lyakh said banned Russian cluster munitions killed four people there - another claim the BBC was unable to verify.
BDST: 1050 HRS, July 3, 2022