The Russian military said on Tuesday that it had enforced a new cease-fire in Ukraine and opened "humanitarian corridors" in five cities.

In the capital Kyiv as well as in the major cities of Chernihiv, Sumy, Kharkiv and the particularly embattled port city of Mariupol, people should be able to get to safety, Ukrainian authorities said Tuesday. Ukraine began evacuating civilians from the northeastern city of Sumy and from the town of Irpin near Kyiv, Ukrainian officials said.

Sumy governor Dmitro Zhivitskiy said in a video statement that the first buses had already departed Sumy for the city of Poltava, further west. He said priority would be given to the disabled, pregnant women and children in orphanages. A short video clip released by presidential advisor Kyrolo Tymoshenko showed a red bus with some civilians on board.

"It has been agreed that the first convoy will start at 10 a.m. (8:00 a.m. GMT) from the city of Sumy. The convoy will be followed by the local population in personal vehicles," Deputy Prime Minister Iryna Vereshchuk said in a televised statement.

Residents were also leaving the town of Irpin, a front-line Kyiv suburb where Reuters journalists had filmed families fleeing for their lives under fierce bombardment on Sunday. Residents ran with their young children in strollers or cradling babies in arms, while others carried pets and plastic bags of belongings.

"The city is almost ruined, and the district where I'm living, it's like there are no houses (that) were not bombed," said one young mother, holding a baby beneath a blanket, while her daughter stood by her side. "Yesterday was the hardest bombing, and the lights and sound is so scary, and the whole building is shaking."

"As of 9:30 a.m. (7:30 a.m. GMT), more than 150 people have been evacuated and (evacuation) activities are underway," said Oleksiy Kuleba, governor of the Kyiv region.

With the invasion well into its second week, Russian troops were making significant advances in southern Ukraine but stalled in some other regions. Soldiers and volunteers fortified the capital, Kyiv, with hundreds of checkpoints and barricades designed to thwart a takeover. A steady rain of shells and rockets fell on other population centers, including the Kyiv suburb of Bucha, where the mayor reported heavy artillery fire.

"We can’t even gather up the bodies because the shelling from heavy weapons doesn’t stop day or night," Mayor Anatol Fedoruk said. "Dogs are pulling apart the bodies on the city streets. It’s a nightmare."

In one of the most desperate cities, the encircled southern port of Mariupol, an estimated 200,000 people – nearly half the population of 430,000 – were hoping to flee, and Red Cross officials waited to hear when a corridor would be established. In Mariupol, this is now the fourth attempt to bring people to safety. They are to be brought out by bus and car. According to the Ukrainian authorities, assembly points are also being set up in the city for this purpose.

The Russian U.N. ambassador forecast a potential cease-fire for the morning and appeared to suggest that humanitarian paths leading away from Kyiv and other cities could give people choice in where they want to go – a change from previous proposals that offered only destinations in Russia or Belarus.

But doubts abounded, fueled by the failure of previous attempts to lead civilians to safety amid the biggest ground war in Europe since World War II. The office of embattled Ukrainian President Volodymyr Zelenskyy would not comment on the latest Russian proposal, saying that Moscow’s plans can be believed only if a safe evacuation begins.

Demands for effective passageways have surged amid intensifying shelling by Russian forces. The steady bombardments, including in some of Ukraine’s most populated regions, have yielded a humanitarian crisis of diminishing food, water and medical supplies. Through it all, Zelenskky said Ukrainian forces were showing unprecedented courage.

"The problem is that for one soldier of Ukraine, we have 10 Russian soldiers, and for one Ukrainian tank, we have 50 Russian tanks," Zelenskyy told ABC News in an interview that aired Monday night. But he noted that the gap in strength was diminishing and that even if Russian forces "come into all our cities," they will be met with an insurgency.

A top U.S. official said multiple countries were discussing whether to provide the warplanes that Zelenskyy has been pleading for.

Mariupol was short on water, food and power, and cellphone networks are down. Stores have been looted as residents search for essential goods. Police moved through the city, advising people to remain in shelters until they heard official messages broadcast over loudspeakers to evacuate.

Hospitals in Mariupol are facing severe shortages of antibiotics and painkillers, and doctors performed some emergency procedures without them. The lack of phone service left anxious citizens approaching strangers to ask if they knew relatives living in other parts of the city and whether they were safe.

In Kyiv, soldiers and volunteers have built hundreds of checkpoints to protect the city of nearly 4 million, often using sandbags, stacked tires and spiked cables. Some barricades looked significant, with heavy concrete slabs and sandbags piled more than two stories high, while others appeared more haphazard, with hundreds of books used to weigh down stacks of tires.

"Every house, every street, every checkpoint, we will fight to the death if necessary," said Mayor Vitali Klitschko, as The Associated Press (AP) reported.

In Kharkiv, Ukraine’s second-largest city, with 1.4 million people, heavy shelling slammed into apartment buildings.

"I think it struck the fourth floor under us," Dmitry Sedorenko said from his Kharkiv hospital bed. "Immediately, everything started burning and falling apart." When the floor collapsed beneath him, he crawled out through the third story, past the bodies of some of his neighbors.

In the small town of Horenka, where shelling reduced one area to ashes and shards of glass, rescuers and residents picked through the ruins as chickens pecked around them.

"What are they doing?" rescue worker Vasyl Oksak asked of the Russian attackers. "There were two little kids and two elderly people living here. Come in and see what they have done."

In the south, Russian forces also continued their offensive in Mykolaiv, opening fire on the Black Sea shipbuilding center of a half-million people, according to Ukraine’s military. Rescuers said they were putting out fires caused by rocket attacks in residential areas.

At The Hague, Ukraine pleaded with the International Court of Justice to order a halt to Russia’s invasion, saying Moscow is committing widespread war crimes.

Russia "is resorting to tactics reminiscent of medieval siege warfare, encircling cities, cutting off escape routes and pounding the civilian population with heavy ordnance," said Jonathan Gimblett, a member of Ukraine’s legal team.

Russia snubbed the court proceedings, leaving its seats in the Great Hall of Justice empty.

Efforts to set up safe passage for civilians over the weekend fell apart amid continued Russian shelling. Before Monday’s talks began, Russia announced a new plan, saying civilians would be allowed to leave Kyiv, Mariupol, Kharkiv and Sumy.

But many of the evacuation routes headed toward Russia or its ally Belarus, which has served as a launchpad for the invasion. Ukraine instead proposed eight routes allowing civilians to travel to western regions of the country where there is no shelling.

Later, Russia’s U.N. ambassador, Vassily Nebenzia, told the U.N. Security Council that Russia would carry out a cease-fire Tuesday morning and made the suggestion that humanitarian corridors leading away from Kyiv, Mariupol, Sumy and Chernihiv could let people choose where they want to seek safety.

The U.N. humanitarian chief, Undersecretary-General Martin Griffiths, addressed the Security Council and urged safe passage for people to go "in the direction they choose."

The battle for Mariupol is crucial because its capture could allow Moscow to establish a land corridor to Crimea, which Russia seized from Ukraine in 2014.

The fighting has sent energy prices surging worldwide and stocks plummeting while threatening the food supply and livelihoods of people around the globe who rely on crops farmed in the fertile Black Sea region.

The U.N. human rights office reported 406 confirmed civilian deaths but said the real number is much higher. The invasion has also sent more than two million people fleeing Ukraine.

At least 21 civilians, including two children, were killed in a Russian airstrike on a residential street in Ukraine's northeastern city of Sumy late on Monday, the regional prosecutor's office said in a statement on Tuesday. The bodies were recovered by emergency services early on Tuesday in searches that are ongoing, it said.

On Monday, Moscow again announced a series of demands to stop the invasion, including that Ukraine recognize Crimea as part of Russia and recognize the eastern regions controlled by Moscow-supported separatist fighters as independent. It also insisted that Ukraine change its constitution to guarantee it won’t join international bodies like NATO and the European Union. Ukraine has already rejected those demands.

Zelenskyy has called for more punitive measures against Russia, including a global boycott of its oil exports, which are key to its economy.

"If (Russia) doesn’t want to abide by civilized rules, then they shouldn’t receive goods and services from civilization," he said in a video address.

World Bank support

The World Bank on Monday approved an additional $489-million package in support for Ukraine, to be made available immediately and dubbed "Financing of Recovery from Economic Emergency in Ukraine," or "FREE Ukraine."

It came as Zelenskyy renewed calls for the West to boycott Russian exports, particularly oil, and to impose a no-fly zone to stop the carnage. NATO countries have so far rebuffed Kyiv's demand for a no-fly zone, fearing a widening war against nuclear-armed Russia.

Western allies have instead imposed unprecedented sanctions against businesses, banks and billionaires in a bid to pressure Moscow to halt its assault. But the leaders of Germany, Britain and the Netherlands warned Monday against a ban on Russian oil, saying it could put Europe's energy security at risk.

U.S. President Joe Biden's spokesperson said no decision had been taken, while Russian Deputy Prime Minister Alexander Novak warned any oil ban would have "catastrophic consequences" on prices that have already headed towards a 2008 record high, according to Agence France-Presse (AFP).

Putin has equated sanctions with a declaration of war and put nuclear forces on alert, pledging the "neutralization" of Ukraine "either through negotiation or through war."

Despite harsh punishments for those voicing dissent, protests in Russia against the Ukraine invasion have continued, with more than 10,000 people arrested since it began.