From what I know, the Pope will only relocate if he has somewhere to do it...

In most cases, when you are playing a Catholic, you'll be asked to give a settlement to the Pope (under threat of Ex-comm, IIRC) and he will re-settle there but otherwise if you're either Orthodox or Muslim, the Pope once gone remains gone forever...

And as you can see in ATPG post, once taken out, the Papal States faction remains gone...