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Ukrainian officials said civilians fleeing the besieged city of Mariupol came under Russian shelling on Tuesday despite the ceasefire agreement, while others in Izium were left stranded after being directed down an escape route covered in landmines, humanitarian agencies reported.
It comes after two attempts to open up routes for civilians failed at the weekend as fighting raged on.
Ukraine's Foreign Ministry spokesman Oleg Nikolenko on Twitter accused the Russian forces of “violating” the humanitarian ceasefire.
Kyiv claims the Russian offer of humanitarian corridors and ceasefires is a publicity stunt and has warned trapped civilians they are unlikely to be able to flee.
“Eight trucks plus 30 buses ready to deliver humanitarian aid to Mariupol and to [evacuate] civilians to Zaporizhzhia. Pressure on Russia MUST step up to make it uphold its commitments,” Mr Nikolenko said.
The UN said on Tuesday that two million people have now fled Ukraine to neighbouring states in Europe’s worst refugee crisis since the Second World War.
Russia on Tuesday again offered to create “humanitarian corridors” and Kyiv appealed to Moscow to allow the guns to fall silent to allow people to flee.
Despite reports of Russian shelling, Ukrainians began fleeing their homes as supplies rushed to the worst-hit areas.
Russia had said it would allow civilians to leave the Ukrainian capital and the cities of Chernihiv, Sumy, Kharkiv and Mariupol as Russian forces in Ukraine introduced a “silent regime” from 7am GMT, the country's defence ministry said.
In the north-eastern city of Sumy and the town of Irpin near the capital of Kyiv, people fled after fighting died down but it was unknown if Ukraine would agree to use other proposed humanitarian corridors that took fleeing people into Russia or Belarus.
But many are sceptical of the Russia-proposed evacuation routes headed towards Russia or allied Belarus, which has served as a launch pad for the invasion. Ukraine instead proposed eight routes allowing civilians to travel to western regions of the country where there is no shelling.
Addressing the Security Council, UN humanitarian chief Martin Griffiths called for safe passage for people to go “in the direction they choose”.
“We have already started the evacuation of civilians from Sumy to Poltava [in central Ukraine], including foreign students,” Ukraine's Foreign Ministry said in a tweet.
“We call on Russia to uphold its ceasefire commitment, to refrain from activities that endanger the lives of people and to allow the delivery of humanitarian aid.”
The evacuation plan drew scepticism from Ukrainian leaders after previous efforts to establish safe passage crumbled at the weekend.
Ukrainian President Volodymyr Zelenskyy's office said Moscow’s proposals can be believed only if a safe evacuation begins.
The steady bombardments, including in some of Ukraine’s most populated regions, have caused a humanitarian crisis, as food, water and medical supplies diminish. More than 400 civilians have been killed since the war began, the UN human rights agency, said although the true number is thought to be significantly higher.
Russian troops have made significant advances in southern Ukraine but have stalled and met resistance in other areas.
Mr Zelenskyy 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,” Mr Zelenskyy told ABC News on Monday night.
But he noted the gap in strength was closing, and if Russian forces “come into all our cities”, they will be met with an insurgency.
Experts have suggested that Russia has a limited ability to maintain the war given the pace of losses so far.
Ukrainian soldiers and volunteers have fortified Kyiv with hundreds of checkpoints and barricades designed to thwart a takeover and to protect the city of about four million people, using sandbags, stacked tyres and spiked cables.
Some barricades look significant, with heavy concrete slabs and sandbags piled more than two storeys high, while others appear more haphazard, with hundreds of books used to weigh down stacks of tyres.
Meanwhile, a steady rain of shells and rockets fell on other population centres, including the Kyiv suburb of Bucha, where its 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 is a nightmare.”
In one of the most desperate cities, the encircled southern port of Mariupol, about 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.
Mariupol is short of water, food and power, and mobile phone networks are down. Shops have been looted as residents search for essential goods.
In Kharkiv, with a population of 1.4 million people, heavy shelling slammed into apartment buildings.
“I think it struck the fourth floor under us,” said Dmitry Sedorenko, a Kharkiv resident wounded in the fighting. “Immediately, everything started burning and falling apart.”
As the floor collapsed beneath him, he managed to crawl out, past the bodies of some of his neighbours.
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 children and two elderly people living here.”
In the south, Russian forces also continued their offensive in Mykolaiv, opening fire on the Black Sea shipbuilding centre of half a million people, Ukraine’s military reported.
Rescuers said they were putting out fires caused by rocket attacks in residential areas.
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.
On Monday, Moscow announced a series of demands to stop the invasion, including that Ukraine recognise Crimea as part of Russia and recognise the eastern regions controlled by Moscow-supported separatist fighters as independent.
It also insisted Ukraine change its constitution to guarantee it will not join international bodies such as Nato and the EU.
Ukraine has rejected those demands.
Mr Zelenskyy 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 civilised rules, then they shouldn’t receive goods and services from civilisation,” he said.
The fighting has sent energy prices surging worldwide and stocks plummeting, while it threatens the food supply and livelihoods of people around the globe who rely on crops farmed in the fertile Black Sea region.
A top US intelligence chief said Russia is likely to face a “persistent and significant insurgency” after Mr Putin misjudged how the war would unfold.
“We judge it will be especially challenging for the Russians to hold and control Ukrainian territory and install a sustainable pro-Russian regime in Kyiv,” she told the House Intelligence Committee on Tuesday during an annual hearing on global threats.
US intelligence agencies assess that Moscow underestimated the strength of Ukraine’s resistance and the degree of military challenges, while Russian forces are operating “with reckless disregard” for the safety of civilians.
Regardless, Mr Putin remains determined to control and dominate Ukraine as he’s been “stewing in a combustible combination of grievance and ambition”, CIA Director William Burns said during the hearing.
Lt Gen Scott Berrier, head of the Defence Intelligence Agency, said he estimated with “low confidence” that Russia has lost 2,000 to 4,000 troops so far in the conflict.