You will have to comply with whatever the local Amex identification policies are. So for example, in Germany I had to submit to Postident to check my passport, which I guess in turn was checked against my Schufa (German credit scoring system).

I think it also depends on the county you are moving to; so for Germany I selected the card I wanted, filled out a new application, and then put my UK card number as an existing card, which then meant that it was handled as a global transfer. Took exactly two weeks from application to receiving card. Other operating countries have their own global transfer pages.

Your card/account doesn't transfer over exactly. You get a new card issued by the country that you have moved to and have the option of cancelling the old card.