It's called "Soli" - "State of License issue" transfer...
I have done this before, it's simple and straight forward, yet expensive (depending on the country you are moving to).
Your current medical will be "converted" in the process-usually, you request the transfer of license with your current CAA and allow them to share your details with the receiving CAA.
The receiving CAA will then receive all documents from your original CAA, and convert the medical in house based on your current medical.
It takes about 2-3 months, and you will have to send your current license to the new CAA before they can issue the new license, as you can only hold one EASA license at any given time..