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The southern Ukrainian port city of Kherson has been over-run by Russian troops.
The mayor says invading forces are on the streets and have forced their way into government buildings.
Other key cities including Mariupol are being hit with sustained shelling, and hundreds are feared dead.
Meanwhile, US officials are warning Russia will increasingly hit civilian targets. And the International Criminal Court has opened an investigation into potential war crimes by Russia.
Tens of thousands of Ukrainians are fleeing over the border to the northern Romanian town of Siret where former journalist and now World Vision aid worker Mike Bruce is based.
The scenes are incredibly sad with a million people now having left Ukraine, the former Cantabrian says.
"We are talking about 400 people an hour crossing that border crossing where we are.
"It is currently snowing; it has been for about three days... it is absolutely freezing, it was minus eight yesterday.
"Yet these people are coming across the border with literally suitcases and backpacks and heading into this..."
Bruce says a lot of Romanians have opened their homes to the Ukrainians but others have been forced into temporary shelters, such as a sports stadium.
"There are literally tents with no heating in the tents so it's pretty miserable, especially in this weather."
Some will stay in Romania; others will move on to other countries such as Hungary or Bosnia or perhaps the EU.
In the first five days after the Russian invasion, 105,000 refugees had entered Romania.
The refugees are mainly women, children and a few older men because most of the men have stayed behind to fight the Russians.
Many women have pushchairs for their young children or are carrying babies wrapped in blankets.
"It's heartbreaking to see what conditions they're coming in, it really is."
Mothers fleeing to countries like Poland and Romania with children wrapped in blankets and very few other possessions. Photo: AFP
He described talking to a woman who was from a western part of Ukraine. Her husband had travelled to eastern Ukraine to join the fighting there.
She told Bruce her heart remained in Ukraine but she felt compelled to escape so she could save her nine-year-old daughter.
Both carried small daypacks - "They're literally just chucking stuff in a bag and going".
He says he has seen many distraught children and World Vision is coordinating 45 psychologists who are offering support.
While food, water and sanitation are important, he says the wellbeing of the displaced children cannot be overestimated.
He says World Vision organised this week for the removal of 40 orphans from Kyiv after their centre was bombed.
They are now in Bucharest, but there are still another 1500 orphans who may need to be rescued.
As well as those leaving the country, many are also displaced within Ukraine abandoning their homes and cities as they seek refuge in other parts of their homeland.
The United Nations estimates that up to 5 million people may flee from Ukraine. At the height of the Syrian crisis 2015-2016 refugees numbered around 1.5m.
"This is the biggest [refugee crisis] since World War II potentially."
He urged New Zealanders to donate to the organisations who were supporting Ukrainians on the ground, including World Vision, and lobby their local MP for a special vote so that the government pays for more humanitarian aid.