try the Living France forums
Depends where in France you are going. Some areas are not as horsey as you might think so finding a good horse vet can be a problem
the availability of compound feeds can be sparse and what there is can be of dubious quality. France still mainly feeds 'straights'
Hay is readily available but they aren't perhaps as ragwort conscious as we are and they to have droughts and suffer shortages
The competition scene is such that you have to pass one of the French riding tests to be able to compete I think. You MUST also check whether your horses will be eligible to compete and/or have their progeny registered in France as their rules are different.
Plus side is that if you are a regd. breeder in France then you should get a breeders premium on all the horses winnings for life so long as it remains in France. You don't get this if the horse is sold abroad.
I'd contact the French National Stud for the area you are moving to and ask their advice. Haras du Pin in Normandy is a good place to start