Don't know about passports. Depends on which agency is producing it. All horses should have them and it is illegal to move them without one.
As regards going to Europe there are lots of companies I'd recommend. Sometimes it will depend where you're starting from and where in Europe you're going. You could try getting quotes from BBA Shipping or Rapido or IRT in Newmarket, or LG Bloodstock/Parkes International in Lambourn area.
I had a good experience with Parkers Int'l, shipped from Norwich to Paris and were very good. Extremely plush horse lorry with cabin video of course and very attentive driver. I was upset that he had removed leg protectors until next time I had to transport my horse and realised that he hates wearing them so very much that he keeps kicking violently until they are removed.
Thank you everyone very much for your replies. Had intended to import Fell mare from Dorset to Belgium.
Disappointingly, since I posted, the sale has fallen through (bad vet's inspection; well perhaps not totally irretrievable but bad enough to make me question vendor's honesty on other matters enough to pull out).
We are looking in Suffolk though as well (sister there) so the Norwich and Newmarket recommendations are much appreciated Butkin and Unobtanium (will file for later use - hopefully - if I ever find the holy grail).
Also, looking for Fell or Highland or Dales so info about obtaining passport through breed society also v. useful too Mitchy