I think it was a tradition in Victorian times and right up until WWI all through Europe. There are pics of Alexei in dresses and ruffles!
In the case of Felix, he said in his memoirs that his mother had hoped for a girl, already having had 3 sons (Nikolai and 2 who had not survived

) She was so sure she was having a daughter she bought all kinds of fancy girls' clothes and shoes to have on hand, and when she had a boy (Felix called it 'my mother's disappointment) she dressed him as a girl anyway until he was 5. He said he didn't mind, he yelled out 'isn't baby pretty!' to those who saw him dressed up in his carriage! Was this the beginning of his love for cross dressing? I wonder what she thought of his dressing in ladies' clothes when he was older! His brother and his girlfriend loved dressing him up, saying he made a pretty girl!