Hi blymer - I would imagine you would have to re-apply for PIP, however don't quote me on that, but I'm pretty sure that you're correct in relation to the UC (not sure with ESA however).
I've approved your post, and maybes someone who's had experience of moving countries can advise you with regards to this query.
Chris.
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
