With milking goats stud owners can either have the female to board (usually only when the female is not milked anymore) for about 3 weeks, to allow for possible return to service, charges are usually around 20pounds for the mating plus any boarding costs (and female owner will usually rbing some of her own feed and hay to allow for a slow changeover to other type food if necessary); or the female is just brought there when she is actually in season, mated there and then (it is usually very quick!) and then returns home - all smelly and happy, and a lot quieter! Stud fees are the same. Most male owners now require a CAE certificate for the female, and the billy should have a valid one too. Stud owner completes a stud certificate, without which you cannot register your kids from that mating, so are quite important. I guess pigmies are the same?
Movement papers need to be completed for both outward and homebound journey, guess its the same in England. Most stud owners have isolation facilities, so any female coming for a quick mating will not cause a standstill.
I have done boarding for my female last year, it worked very well - but you have to be comfortable with the studowners set-up, how s/he handles your goats and that the right billy will mate your girl! This year it was drive-by matings, with one return journey, again it worked alright.
I had to de-louse my goats just recently and it seems the only place they could have picked some passengers up was during matings away. It's not a problem, but maybe worth keeping in mind (and have some Spot-on at hand).