I would have hoped that when you renew your JAA Class 1 or 2 medical then you will automatically get the EASA version, i.e. nothing actually changes and there isn't anything you need to do.
Is that correct?
AFAIK EASA is nothing nothing of great significance on the medical front. They are retaining the stupid Class 1 audiogram for the IR.
There is some fun stuff regarding renewing a medical in a different country. The foreign AME will need to be briefed by your home AME. No doubt there will be a charge for that
But it is to discourage medical tourism.