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They are so scared. what are they telling you? >> reporter: they're telling me they're exhausted, overwhelmed and desperate. the numbers are staggering. over 670,000 and almost two-thirds of those, according to polish authorities, are ending up in poland and many transiting through lviv. the western most train hub here in ukraine. it is almost impossible to board a train out of the country unless you are a woman or a child and even for them, it is extremely difficult. we have seen countless scenes of desperation, of people trying to board trains in recent days and even when they arrive on the ...

Train station and talks with those whom the city promised of escape. >> reporter: with each new round of bombing and shelling comes a new wave of people seeking refuge outside ukraine. many arrived in the western train hub of lviv, people lined up outside the doors of the station for the next train to poland. those who wait are almost entirely women, girls and boys who suddenly look a lot like men. this family fled central ukraine, yulia left her murkts father and brother behind to fight the russians now going to poland with her mother and two boys aged 1 and 16. >> when my husband left he sai...

Desperation only grows for those hoping to escape the violence. cnn's scott mcclain reports now from a train station in lviv. >> reporter: with each new round of bombing and shelling comes a new wave of people seeking refuge outside ukraine. many arrive in the western train hub of lviv where people are lined up out the doors of the station for the next train to poland. those who wait are almost entirely women, girls and boys who suddenly look a lot like men. this family fled central ukraine. she left her brother, father and husband behind to fight the russians. now she's going to poland with h...

Covering this for us. he's in a train in lviv this morning. scott, what have you seen there today versus what you've seen in days past? >> reporter: hey, erica, the chaos is getting a heck of a lot more organized. on the borders, we're hearing it's getting much easier to cross, but there are still a marked number of people who want to go by train, especially if you have small kids. we're standing on the platform level. they're not letting anyone up here. that's because everyone is lined up down here at the bottom. and just to clarify, they are all waiting for trains onward into poland. this is...

As they try to evacuate because the mayor of lviv, as you know, has now said that lviv needs -- it's almost reached capacity for their ability to absorb this influx of people fleeing, that they need tents, medical supplies, supplies to take care and deal with just the tens of thousands, hundreds of thousands of people who have been coming through. >> yeah, exactly. and lviv is a western train hub. all the trains from kharkiv, all the trains from kyiv, many of them end up in lviv. from there, people have to get off, wait in line again. lines are stretching out of the station. and then wait to b...