Apologies if I’ve missed this in a very obvious place, but once the Ambas have built and then swap their PI for a mine, does it go back onto its space on their board, and then do they need to rebuild it
No, you did not miss anything and if that would be the case, it sure would have been spelled out in the rulebook.
or do can they simply swap it again on another turn? Do they also retain the power gain/charge income?
Yes to both of those.