What sort of cycle are you on? Sometimes they may want to do a blood test a few days after the transfer to make sure your hormone levels are ok (with HRT cycle or maybe if you need extra progesterone support).
I think the flying itself would be fine (a holiday's a good idea even), the only issue would be if you need any follow up testing or care through your clinic. Probably best to ask them.

I *think* your period would be due about the same time as usual, but it depends again on what sort of cycle you're doing. If you're on progesterone, for eg, you may not get a bleed till you stop it.

Good luck!