"...there's a clear correspondence of names getting big in the UK and then getting big in the US 5-10 years later."
Or vice versa as in the case of
Edith and maybe
Florence, which were popular in the US and then caught on in Britain. The usage of some names like
Alice and
Evelyn don't seem to have been inspired by another countries use; it seems more related to old fashioned names making a comeback.
With names like
Sophia, I think the US trend may have been more inspired by Italy and widespread trends, and
Camila is more due to Spanish influence.