Ukrainian authorities on Monday ordered families with children to urgently evacuate the eastern city of Pokrovsk, where the Russian army is bearing down after a six-month slog across Ukraine's Donetsk region following the capture of Avdiivka.

Local authorities said Russian forces are advancing so quickly that families must leave the city and other nearby towns and villages. Some 53,000 people still live in Pokrovsk, officials say.

The relentless assault on Ukrainian strongholds in Donetsk and neighbouring Luhansk, which together make up the Donbas industrial region, has cost Russia heavily in troops and armour.

However, the onslaught has gradually paid dividends as Ukrainian defenders have no choice but to pull back from positions blown to pieces by Russian artillery, missiles and bombs.

One of Kyiv's attempts to ease the pressure was the August 6 incursion into Russia's Kursk region, opening an unexpected front in the hopes of unnerving the Kremlin and compelling it to split its military resources.

Ukrainian President Volodymyr Zelensky said on Sunday the daring incursion is trying to create a buffer zone that might prevent further attacks by Moscow across the border, especially with long-range artillery, missiles and glide bombs.

That Ukrainian operation was continuing on Monday under tight secrecy.

In Pokrovsk residents have just two weeks to leave the city safely, officials said in an interview with the US-funded Radio Liberty. Officials warned last week that Russian forces were rapidly advancing and were just 10 km from the city's outskirts.