Ukrainian ­authorities vowed Saturday to restore control over the roiling eastern part of their nation, slowly advancing on two key breakaway cities even as the Kremlin and its supporters in Ukraine said the violence demanded a response.

The military operations Saturday claimed at least 10 lives, medical officials said, a day after a conflagration in a trade union building killed ­dozens of pro-Russian activists in the port city of Odessa in the bloodiest day in Ukraine in nearly three months.

Ukraine’s acting president, ­Oleksandr Turchynov, declared a two-day national mourning period, as grieving residents of Odessa streamed to the burned-out building to lay flowers. Thousands of mourners took to the streets of the city, which until the previous day had been largely untouched by the kind of violence that has plagued eastern Ukraine for weeks.

The violence was an ominous development in the unfolding turmoil in Ukraine, because Odessa, more than 300 miles to the west of the bulk of separatist sentiments, lies on a belt extending into Moldova that Russian President Vladimir Putin has noted was historically part of the Russian Empire.

Reporters in Odessa said the mourners chanted, “Odessa is a Russian city.”

Map: Russia and Ukraine are positioning their troops for war.

By day’s end, Ukrainian authorities in the east said they had encircled Slovyansk and retaken portions of Kramatorsk, including a television tower that had been seized by pro-Russian activists, although the core of the city appeared to remain in rebel hands. Separatists in Slovyansk on Saturday freed a team of international observers who had been held hostage for more than a week, after Putin sent an envoy to push for their release.

For every advance that the Ukrainian government made, it seemed to lose ground elsewhere. Angry pro-Russian crowds seized control of more government buildings in Donetsk, and pro-Russian forces in Luhansk, a city just 15 miles from the Russian frontier, vowed war on Kiev, declaring a curfew and seizing weapons inside a military recruitment center.

The spate of violence prompted the Kremlin to warn that it was weighing how to respond as thousands of Russian troops massed along the border. Putin has previously said he would be prepared to intervene if the interests of compatriots in Ukraine were under threat.

“People are calling in despair, asking for help. The overwhelming majority demand Russian help,” Kremlin spokesman Dmitri Peskov told reporters Saturday. “Thousands” of calls had come in the previous 24 hours, he said, although he offered no evidence.

“All these calls are reported to Vladimir Putin,” he said. He blamed Kiev and its European and U.S. supporters for the violence, saying that “their arms are up to the elbows in blood.”

In Kiev, authorities said that they would not relent in their efforts to repel separatists in eastern Ukraine but that police officers in Odessa might be held accountable for allowing the violence there. On Friday, a pro-Ukrainian rally in the city attended by thousands of soccer fans was attacked by pro-Russian separatists, sparking hours of street battles and causing the deaths of three people. Later that evening, a pro-Ukrainian mob attacked people in a pro-Russian encampment, sending them running into a nearby building that the mob then set on fire with gasoline bombs. Many people were trapped inside.

At least 46 people died in the clashes, almost all of them in the fire, authorities said Saturday.

Kerry: Release is ‘a step’

Russian envoy Vladimir Lukin, Donetsk Gov. Serhiy Taruta and Thorbjorn Jagland, a Norwegian politician serving as secretary general of the Council of Europe, confirmed the OSCE hostages’ release at a news conference Saturday, saying it had been secured without conditions. They said there was no sign that the hostages, European military observers, had been tortured or mistreated.

The OSCE hostages “had a very good attitude, and that gave them the strength to stand the situation,” said German Col. Axel Schneider, one of the detainees, the Associated Press reported. “We have been treated as good as possible.”

Secretary of State John F. Kerry welcomed the release of the OSCE personnel and condemned violence “by any side.”

“It’s a step,” he said of the release. “But there are many other steps that need to be taken in order to be able to de-escalate the situation.”

Kerry, speaking in Kinshasa, Congo, said he had discussed those steps in a call Saturday to Russian Foreign Minister Sergei Lavrov. He said he reiterated the warning of economic sanctions against Russia issued by President Obama and German Chancellor Angela Merkel this past week, adding that any such measures would include broad sanctions on sectors of the Russian economy. He did not elaborate.

He said he and Lavrov also discussed the rising level of violence in eastern Ukraine.

“The United States condemns the violence that has been taking place by any side,” Kerry said. “That includes the violence of anyone who lit a fire and caused the deaths of those 38 people or more in a building in Odessa.”

But the two officials appeared to be talking past each other.

Lavrov told Kerry that “the punitive operation in southeastern Ukraine is putting the country into a fratricidal conflict,” the Russian Foreign Ministry said in a statement after the conversation. Russia’s top diplomat called on the United States “to use all its influence to force the Kiev regime protected by it, which has declared a war on its own people, to immediately halt the military action in the southeastern regions, remove the troops and release protesters.”

Lavrov also called for the OSCE to help de-escalate tensions and establish a “nationwide dialogue” focused on constitutional changes that Russia hopes would give more autonomy to the ethnically Russian eastern parts of Ukraine, a measure that would award Russia more influence over the region.

But, Lavrov said, “chances still remain” for a dialogue.

Looting in Donetsk

On the ground in eastern Ukraine, Kiev’s control Saturday appeared patchy. In Donetsk, masked men, some dressed in black, others in camouflage, smashed and looted two buildings that symbolized the Ukrainian state as hundreds of civilians chanted, “Odessa will not be forgiven,” as well as “Russia” and “No to fascism.”

The men attacked the abandoned office of the State Security Service and the office of a metallurgical company partly owned by Taruta, the Kiev-appointed governor. They smashed windows with batons and sticks, and carried away computers, keyboards, bottles of wine and spirits — almost anything they could lay their hands on, including an office chair and several paintings. Ironically, the company is also partly owned by Russian businessmen.

Papers fluttered from broken second-floor windows, and the crowd cheered as a man flew the Russian flag from a balcony and when a masked man in black stood on the steps outside the building with the black, blue and red flag of the self-styled Donetsk People’s Republic.

Earlier, thousands gathered outside the headquarters of the separatist republic to hear leader Denis Pushilin denounce the deaths in Odessa as unimaginable “on our land, on Russian land.”

In the crowd, one man said Soviet dictator Joseph Stalin should be taken from his grave to help fight the “Nazis,” referring to pro-Ukrainian forces. A woman pointed at a Western journalist and threatened to throttle him with her bare hands if he was an American, blaming the United States for sending mercenaries to support the Ukrainian army.

Birnbaum reported from Moscow. Anna Nemtsova in Luhansk, Ukraine, Alex Ryabchyn in Donetsk, Ukraine, and Anne Gearan in Kinshasa, Congo, contributed to this report.