It is true that the competent authority holds the medical records and, if the pilot requests to change the competent authority, they are transferred along with the licensing records (see FCL.015(d)), but this does not prevent a pilot obtaining a medical certificate in another state. A Brit living in France with a UK licence can visit a French AME who will issue the certificate and send the record of the medical examination to the UK CAA. It's no different to revalidating a class or type rating in a different state.
Got distracted while writing the above and didn't see 421C's last post, which says much the same thing.