Two civilians were killed on Friday in shelling on Russia's Belgorod region on the border with Ukraine, while Moscow's forces attacked Kyiv for a sixth day in a row.
Over the past few days, thousands of residents have fled villages near Russia's southwestern border with Ukraine as shelling has intensified.
Residents from the town of Shebekino, which has been hard hit, have poured into the region's main city, also called Belgorod.
At Belgorod's sports arena, which has become the city's biggest centre for displaced people, AFP journalists saw volunteers organising the transfer of displaced people to smaller dormitories.
A dozen people huddled in front of a TV, and some grew emotional as they watched news developments from Shebekino.
Belgorod governor Vyacheslav Gladkov said 2,500 people were being accommodated in temporary shelters, including the sports arena.
"We have never had a situation like this," Gladkov told reporters at the arena.
He said the conditions at the arena were becoming "cramped" and displaced people were therefore being moved to other spots.
Belgorod mayor Valentin Demidov told AFP that a total of 5,000 people had transited through the temporary shelters and registered in recent days, with many then going to stay with relatives.
"We are trying to re-settle people as quickly as possible," he said.
"We hope they will not have to be there (in temporary accommodation) for long but we have to prepare for anything," he added.
- 'Died on the spot' -
Earlier in the day, the regional governor said two women were killed by shelling.
"Shrapnel hit cars passing by. Two women travelling in one of them died on the spot from their wounds," he said.
He later said a total of 21 people had been injured in four areas in the region.
The attack "involved up to 70 militants, five tanks, four armoured vehicles, seven pickup trucks and a Kamaz truck," the army said.
Following the uptick in violence on the border with Ukraine, the Kremlin has denounced the silence from the international community despite "every opportunity to see the footage describing strikes on residential buildings, social infrastructure."
The United States has said it does not support attacks inside Russia and is instead providing Kyiv with equipment and training to defend itself.
Ukraine's air defence has been boosted by Western equipment, including Patriot air defence systems, seen as one of the most advanced US air defence systems.
- Air strikes on Kyiv -
The Ukrainian capital has seen almost nightly air raids in May after being relatively spared in the beginning of the year.
On Friday, Ukraine said it had destroyed all 15 missiles and 21 drones from a new wave of overnight attacks that left two people wounded in Kyiv.
Russian shelling also killed two women and wounded another four people in the southern region of Zaporizhzhia, Ukrainian officials said.
The Russian defence ministry said forces struck Ukraine's air defence systems protecting "key" military infrastructure and that "all assigned targets have been hit."
The intensified strikes on Kyiv, far from the frontlines in the east and south of the Western-backed country, have raised questions about Russia's calculus.
Some experts say the Kremlin might be seeking to hinder Ukraine's planned counter-offensive and deplete its air defences.
- 'Knockout' -
On Thursday, three people including a nine-year-old child were killed in Kyiv as a result of falling rocket fragments when Russia pounded Kyiv with ballistic and cruise missiles.
The husband of one of the victims, Yaroslav Ryabchuk, said the shelter where they routinely hid from Russian strikes was closed.
Following the latest tragedy, Ukrainian President Volodymyr Zelensky said that all shelters in Kyiv must be inspected.
Klitschko said measures had been taken to ensure "round-the-clock access to shelters".
"The enemy is now shelling the capital with ballistic missiles," Klitschko told residents, saying they might only have "a few minutes" to hide.
He advised residents to shelter in their homes, with two walls of protection from the outside, "if you understand that you won't reach the shelter that quickly".
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© Agence France-Presse
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