The United Nations estimates the number of people fleeing the violence has now passed one million.
Officials say more than half have gone to neighbouring Poland - where families are seen carrying belongings in carrier backs on packed out trains.
At the Romanian border, some women and children are in tears having left their homes.
Eleanor Monbiot is from Christian charity World Vision which is helping support refugees at the border.
She tells Premier they are now seeing unaccompanied children arriving: “In a matter of days, we have children who have lost their homes, who have had their families torn apart, because their fathers are having to stay in Ukraine.
"They have become refugees. They've had to shelter from bombs.
“They're now in a country where they don't speak the language and cannot make themselves understood.
"So we're seeing really traumatised, shaken scared mothers with children at the borders.
“The immigration checks are very slow so it's taking between 20 and 60 hours to cross the border.
"It’s below freezing and it’s snowing right now. So they're exhausted. They're very, very emotional. This is an incredibly hard time.
“Already, we're seeing unaccompanied children appearing at the borders, or they've maybe travelled within non-family groups, but it makes them incredibly vulnerable.
“Just this morning, we were speaking to Timothy, he's 17.
"His grandfather walked with him across the snow to the border, dropped him off and returned.
"Now Timothy is trying to make his way on his own to Italy and he is in tears, he is in pieces.
“He is on his own for the first time in his life. He's left his family, his friends, he's terrified.
"He's also unfortunately, one of the many thousands and possibly even higher number of children that we'll see who are now particularly vulnerable to abuse.
“We do a lot of work in the Balkans, with unaccompanied migrants from Afghanistan, from Iraq, from other countries.
"The stories will break your heart of what happens to these children between when they leave their country and when they come to us and even when they make it to the borders of the EU, they're still not safe.
“We are working over the years with anti sex trafficking programmes, because these children are picked up and brought into the EU to go into sex rings and this is what we know can happen to the unaccompanied children in Ukraine. Unless we put in the right processes and procedures right now.
“Only God can see us through this. Only prayer can break this and we really need to come together as a family of Christians.”
World Vision is asking people to pray for peace in Ukraine.
#PrayForTimothy