Your medical records will be held in the state in which you had the first medical (providing they haven’t been SOLI’d (transferred)). Thereafter, renewals can be done in any EASA country by any EASA Doctor and the results sent back to your original state.
That may be true within EASA theory/regulation but how? What states are accepting medical renewals undertaken in another state and what are the protocols? Non UK flight examiners can register with the UK CAA, having completed an on-line course, to examine UK pilots. It is news to me (which would not be surprising in the current chaos of things) that similar arrangements are in place for AMEs because there are problems of privacy.
A major stumbling block has always been the sharing of confidential medical records in most if not all states. The Germans have even blocked SOLI involving them because German law prevents the exchange of medical data.