Ukraine has transported at least 3,000 people from the battled-scarred eastern region of Donetsk since it initiated mandatory evacuations ahead of gruelling winter months, Kyiv said Tuesday.

"In the last six days, more than 3,000 citizens were evacuated including almost 600 children and 1,400 women. Obligatory evacuations continue," said Kyrylo Timoshenko, the deputy head of the Ukrainian president's office.

Kyiv said it had launched compulsory evacuations last week from the eastern Donetsk region -- now bearing the brunt of Russia's offensive -- because the government does not expect to be able to provide it with heat in the cold winter months.

Timoshenko said that since Russia invaded in February around 1.3 million people were evacuated from Donetsk, which has been under the partial control of Russian-backed separatists since 2014.

"Now on the territory of the region there is a population of 350,000, including 50,000 children," he added.

Many Ukrainians in the eastern Donbas region -- comprised of Donetsk and Lugansk -- have been reluctant to leave, citing a lack of resources to start a new life or not having anywhere else to go.

Donetsk and Lugansk have weathered the worst of the fighting and Moscow's forces have captured most of the cities in the two war-torn regions.

The Ukraine presidency said Tuesday morning that over the last 24 hours three people had been killed and 19 more injured in Russian bombardments across the entire region.

Ukrainian President Volodymyr Zelensky late last month urged people to heed the orders, saying more people would survive the war if they fled Russia's advance.