According to TASS news agency, three people had been killed and 11 injured in Ukrainian shelling of Donetsk.
In a statement posted on Telegram, Denis Pushilin, the head of those parts of Donetsk region that Russian controls, said the casualties occurred when Ukrainian shells hit a civilian bus.
Russian-installed Donetsk Mayor Alexei Kulemzin separately published on Telegram pictures of a burnt-out bus.
Later on Monday Yevgeny Balitsky, the Russian-installed acting governor of Zaporizhzhia region, said three people were killed and about 15 injured after Ukrainian shelling of a local store in the village of Basan.
A Russian missile strike on the southern Ukrainian city of Kryvyi Rih on Monday killed at least six people, including a 10-year-old girl and her mother, and wounded dozens of others, Ukrainian officials said.
A video posted by President Volodymyr Zelenskiy showed smoke billowing from a gaping hole smashed in the side of a nine-storey residential building, and another four-storey building almost levelled.
"It's already six dead in Kryvyi Rih," regional governor Serhiy Lysak wrote on Telegram, announcing a day of mourning in the city.
Mayor Oleksandr Vilkul said a 10-year-old girl and her 45-year-old mother were among the dead.
Ievhenia, a city resident, said two missiles flew over her home just before 9am and she rushed to the scene of the impact after her friend who lives nearby called her.
"What we saw was pure horror, committed by the Russians... They hit a residential building, and her block is just next to it, so everything in their flat was ruined," she said.
Lysak said 75 people had been wounded, including six children aged from four to 17-years-old.
Nearly 150 of the building's residents managed to get out by themselves and 30 were helped out by rescuers, he added.
"Think about it, how many people! And they launched a missile at them," he said.
Vasyl, a resident whose home was damaged in the strike, said he had moved to Kryvyi Rih from the frontline region of Kherson in the south.
"I don't know how I'll fix the windows. I'm an internally displaced person from Kherson region, and I came here, and the same things are starting to happen."
Zelenskiy, who grew up in the steel-producing city with a pre-war population of more than 600,000, said the strikes had hit a residential building and a university building.
"This terror will not frighten us or break us. We are working and saving our people," he said on Telegram.
More than 350 rescuers were involved in an operation to save people trapped under the rubble, Zelenskiy said.
Air force spokesman Yuriy Ihnat said the attack appeared to have been carried out with ballistic missiles.
"This is already a kind of genocidal everyday reality... Kryvyi Rih," Zelenskiy's adviser Mykhailo Podolyak said on Twitter.
"International law will never work if the aggressor does not see a real power behind it. The power begins with closing the Ukrainian skies with missile defence and air defence systems," he said.
The United States and European countries have supplied sophisticated air defence systems that have largely protected places like the capital Kyiv from wave after wave of attacks.
The sheer size of Ukraine makes it hard to ensure the same level of defences across the country, and ballistic missiles are also particularly difficult to shoot down.
