Generally it is not a good idea to change your State of Issue just because you are training in a different State. If you are going to live & work there, then fine go ahead and change.
The medical records have to be kept by the State of Issue.
If you are training in a different state, then the Flight Examiner must be briefed by the state of licence issue. There is a document on the EASA website called Examiner Difference Docment
The European Aviation Safety Agency (EASA) today published the Examiner Differences Document | EASA To cover this.