I have flown with horses numerous times with Dutta Corp. The cost will depend on what size slot in the horse box you select. Three can fit and like a trailer the dividers are adjustable. (plus the last time I flew with horses was 10 yrs ago my pricing is not current).
Personally, I would only ship with the three stall confiration (depending on the size of the horse) because they tend to lean during take off and landing if they are not used to traveling.
Just call Dutta and see. You know you would be flying out of a major airport in either GB or UK and into JFK.
They ask for the stable location because they also can arrange transport to the airport.
My guess, at least $5-7k to start.
Look up Julie Burns Richards Eventing. She was on the US team for eventing and imports Irish Sport horses. Fernhill horses. She may have something stateside.