A Russian border town becomes a front line in the Kremlin’s war in Ukraine.

Kyiv’s mayor, Vitali Klitschko, said on the Telegram messaging app that an additional 16 people were injured by debris from air defense systems shooting down incoming attacks. Ukraine’s general staff headquarters said Kyiv had been attacked by a volley of 10 Iskander ballistic missiles, all of which were shot down.

The city’s military administration said some of the debris fell on a clinic and an adjacent building. A mother and child were killed minutes after the air-raid alert while trying to get into a shelter at the clinic, according to Mr. Klitschko and two emergency workers at the scene who declined to be named because they were not authorized to speak to the news media.

The mother and child who died were not immediately identified.

A man who lives near the clinic and who gave his name as Yaroslav told the Ukrainian public broadcaster Suspilne that his 33-year-old wife, Natalya, was also killed after she and their 9-year-old daughter, Polina, were unable to get into the shelter.

Yaroslav said people were trying desperately to get in. “People were knocking, knocking for a very long time,” he said, adding, “There were women and children and nobody opened it.”

After what he described as an explosion, Yaroslav found Natalya bleeding alongside a blanket she brought for their daughter and a blue bag carrying family documents, according to the Suspilne report. His daughter, who was not injured, saw what happened to her mother, Yaroslav said.

City officials opened a criminal investigation into the clinic and the administrators responsible for operating the shelter there, centered on whether the shelter was properly maintained and why it may have been inaccessible, Mr. Klitschko said. Police officers will now patrol bomb shelters during air raids to make sure they are open, he added.

“Now the investigation is establishing whether the shelter was open,” he said on Telegram.

It was yet another night that Kyiv’s 3.6 million residents were jolted out of bed and sent scurrying for cover. The overnight attack early Thursday left very little time for residents to take shelter, with air defenses colliding with missiles six minutes after the alarm sounded, the Kyiv military administration said.

Throughout May, residents were dogged by 17 waves of aerial attacks coming at all hours, including assaults with drones and ballistic and hypersonic missiles.

While Kyiv has been attacked since the first days of the war, the pace and intensity of the Russian assaults over the past month have been jarring even for civilians now accustomed to spending hours in bomb shelters and sleepless nights huddled in corridors. Thursday’s strikes seemed to suggest that the campaign would continue into June.

On Wednesday, in a speech commemorating International Children’s Day, President Volodymyr Zelensky of Ukraine said that at least 483 Ukrainian children had been killed since the start of Russia’s full-scale invasion, and that untold others had had their rights to a safe environment, education and health care violated by Russia’s invasion. Other estimates say the number of children killed may be even higher.

“For 15 months, Russian aggression and terror have been destroying not just buildings, but fundamental human rights — the fundamental rights of our children,” he said.

Officials in Kyiv said that some Children’s Day events scheduled for Thursday had been canceled. The day “has to be about safe childhood, summer, life…,” the Ukrainian first lady, Olena Zelenska, said on Twitter. “But today it is about new crimes” of the Russian Federation against children, she added.

The week started with a rare daytime attack on the Ukrainian capital, when missiles roared into Kyiv shortly after 11 a.m. Monday and sent schoolchildren running in fear. Every missile was intercepted by the air defense systems, but debris caused fires and other damage.

Russian forces have changed the timing of bombardments, the combination of weapons used and the trajectories of missiles and drones, lately flying them low along riverbeds and through valleys to avoid detection, Ukrainian officials say.

Andrew E. Kramer and Nicole Tung reported from Kyiv, Ukraine, and Victoria Kim from Seoul. Marc Santora contributed reporting from Kyiv, and Juston Jones and Anushka Patil from New York. Dmitriy Khavin contributed translation.