We could draw a parallel with Prince and Princess Mikhail Cantacuzene, who saw the writing on the wall in the summer of 1917 and sent their three children out of the country. Princess C was American (a granddaughter of President Grant), and their 16-year-old son and two rather younger daughters went to San Francisco via the Trans-Siberian Railway and a ship from Vladivostock, accompanied by a tutor and a couple of servants. It is clear from Princess C's account that the parents had very mixed feelings, but decided that it was for the best. The journey went without a hitch - the difficult part, apparently, was getting all the documentation beforehand.
I think the Provisional Government might well have allowed the girls to depart (though possibly not Alexei, since he was the heir), and finding a country prepared to take them would have been less difficult than it was in the case of Nicholas and Alexandra. Of course, the girls themselves may have insisted on staying!
Ann