Buses packed with people fleeing the Russian invasion in Ukraine have left two embattled cities along safe corridors, while officials said the exodus of refugees from the country has reached million.

The Russian onslaught has trapped people inside besieged cities that are running low on food, water and medicine amid the biggest ground war in Europe since the Second World War.

Previous attempts to lead civilians to safety have crumbled with renewed attacks, but on Tuesday, video posted by Ukrainian officials showed buses packed with people moving along a snowy road from the eastern city of Sumy and yellow buses with a red cross on them in southern port of Mariupol.

It was not clear how long the effort would last.

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Displaced Ukrainians queue to board a bus to Poland (Bernat Armangue/AP)

“The Ukrainian city of Sumy was given a green corridor, the first stage of evacuation began,” the Ukrainian state communications agency tweeted.

The buses are heading to other cities in Ukraine, but many people have chosen to flee the country. Safa Msehli, a spokeswoman for the UN’s International Organisation for Migration, tweeted that two million have now fled the country, including at least 100,000 who are not Ukrainian.

Ukrainian President Volodymyr Zelensky called for the expansion of humanitarian corridors for civilians fleeing the war.

In a video address on Tuesday from an undisclosed location, he said a child had died of dehydration in Mariupol, in a sign of how desperate the city’s population has become, and pleaded again with western countries to provide air support.

He said evacuation buses have been sent to Mariupol, but there was no firm agreement on the route, so “Russian troops can simply shoot on this transport on the way”.

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Refugees on a bus from the Mariupol area (AP)

With the invasion well into its second week, Russian troops have made significant advances in southern Ukraine but stalled in some other regions.

Ukrainian 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 centres, 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,” Anatol Fedoruk said. “Dogs are pulling apart the bodies on the city streets. It’s a nightmare.”

In Mariupol, an estimated 200,000 people — nearly half the population of 430,000 — hoped to flee.

Russia’s co-ordination centre for humanitarian efforts in Ukraine and Ukrainian deputy prime minister Iryna Vereshchuk both said a ceasefire had been agreed to start on Tuesday morning to allow some civilians to leave, but it was not clear where all the corridors would lead to, amid disagreement between the two sides.

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(PA Graphics)

Russia’s co-ordination centre suggested there would be more than one corridor, but that most would lead to Russia, either directly or through Belarus. At the UN, the Russian ambassador suggested corridors from several cities could be opened and people could choose which direction to take.

Ms Vereshchuk only said that the two sides had agreed an evacuation of civilians from Sumy, towards the Ukrainian city of Poltava. Those to be evacuated include foreign students from India and China, she said.

She reiterated that proposals to evacuate civilians to Russia and its ally Belarus, which was a launch pad for the invasion, were unacceptable.

Later, Ukrainian presidential aide Kyrylo Tymoshenko posted a video of yellow buses with a red cross on the side that he said showed evacuations from Mariupol toward the city of Zaporizhzhia. He said that humanitarian aid was also being sent to Mariupol.

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.

Mr Zelensky 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,” he told ABC News in an interview that aired on Monday night. But he noted that the gap in strength was closing and that even if Russian forces “come into all our cities”, they will be met with an insurgency.

A senior US official said multiple countries were discussing whether to provide the warplanes Mr Zelensky has been pleading for.

Ukrainian defence minister Oleksiy Reznikov said at least 400 civilian deaths have been recorded and 800 have been wounded, but added that the figures are “definitely incomplete”.

In a video address, he added that Russian military actions have killed 38 children and wounded more than 70, and destroyed more than 200 Ukrainian schools, 34 hospitals and 1,500 residential buildings.

Mariupol was short on water, food and power, and mobile phone 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.

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Volodymyr Zelenskyy (Ukrainian Presidential Press Office/AP)

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.

Several hundred miles west of Mariupol, Russian forces continued their offensive in Mykolaiv, opening fire on the Black Sea shipbuilding centre of half a million people, according to Ukraine’s military. Rescuers said they were putting out fires caused by rocket attacks in residential areas.

Ukraine’s general staff of the armed forces said in a statement that Ukrainian forces are continuing defence operations in the suburbs of the city.

The general staff said “demoralised” Russian forces are engaging in looting in places they have occupied, commandeering civilian buildings like farm hangars for military equipment, and are setting up firing positions in populated areas.

Ukrainian defence forces were also involved in operations in the northern city of Chernihiv and the outskirts of Kyiv, the general staff said.

In Kyiv, soldiers and volunteers have built hundreds of checkpoints to protect the city of nearly four million, often using sandbags, stacked tyres and spiked cables.

Some barricades looked significant, with heavy concrete slabs and sandbags piled more than two storeys high, while others appeared more haphazard, with hundreds of books used to weigh down stacks of tyres.

“Every house, every street, every checkpoint, we will fight to the death if necessary,” said Kyiv mayor Vitali Klitschko.

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.