Four civilians were killed on Friday in shelling on Russia's Belgorod region bordering Ukraine, a local official said, 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 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 center for displaced people, AFP journalists saw volunteers organizing the transfer of displaced people to smaller dormitories.
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 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.
'Won't drag NATO into war'
Earlier in the day, the regional governor said two women were killed by shelling.
"Shrapnel hit cars passing by. Two women traveling in one of them died on the spot from their wounds," he said.
He later said that two other civilians were killed in a rocket attack on the village of Sobolevka, while over 20 people, including children, were injured throughout the region during the day.
On Thursday, the Russian Defense Ministry said forces had used jets and artillery to repel a Ukrainian attempt to "invade" the region of Belgorod.
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 defense has been boosted by Western equipment, including Patriot air defense systems, seen as one of the most advanced U.S. air defense systems.
However Ukraine's President Volodymyr Zelensky acknowledged on Friday that Ukraine would not be able to join NATO before the end of the Russian invasion.
"We are reasonable people and we understand that we are not going to drag a single NATO country into a war," Zelensky said during a briefing along Estonian President Alar Karis.
"Therefore, we understand that we will not be members of NATO while this war is going on. Not because we don't want to, but because it's impossible," Zelensky added.
Ukraine is a candidate to join both NATO and the European Union, but some European capitals are wary of setting a formal timeline for membership as Russia's invasion continues.
Joining NATO would mean Ukraine would be covered by the alliance's Article 5 collective defense clause that obliges all members to help defend it if attacked.
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.
"In the last six days, [Russians] have already carried out six attacks on the city!" the chief of the capital's city administration, Sergiy Popko, said on Telegram.
Russian shelling also killed two women and wounded another four people in the southern region of Zaporizhzhia, Ukrainian officials said.
In the southern region of Kherson two children were injured in a playground by an explosion, a local official said.
The Russian Defense Ministry said forces struck Ukraine's air defense systems protecting "key" military infrastructure and that "all assigned targets have been hit."
'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, Zelensky said that all shelters in Kyiv must be inspected.
On Thursday, he vowed that all those responsible would be held accountable, saying "there may be a knockout," apparently a thinly-veiled warning directed against Kyiv Mayor Vitali Klitschko, a former boxer.
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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