The question is too general, since different coutnries may have different permanent residence permits (both regulated by EU or local laws), therefore my answer is first of all related to Germany (since the question mentions it as an example).
In such cases, I would (and actually I have already did it) get 2 permanent residence permits in Germany first:
Niederalssungserlaubnis (§9 AufenthG or §18c AufenthG or maybe some other §)
Daueraufenthalt-EU (§9a AufenthG)
Then, it might be possible to keep the first one for Germany (one may need to take into account the expiration policy) and exchange the second one for another country (here it very much depends on the laws of that other country).
This way might be still complicated, especially when Niederlassungserlaubnis Blaue Karte may be granted after 21 months working in Germany, but for Daueraufenthalt-EU one needs to live 5 years in Germany.
IMHO, if one has non-european permanent residence permit (Niederlassungserlaubnis) and lived less that 5 years in Germany, such kind of permanent residence on its own gives almost no preference for obtaining permanent residence in another EU country. E.g., Germany itself only gives preference for holders of European permanent residence permit.
Other option would be just to get any European citizenship and then live in any EU country using the EU rights of free movement. The logic is that if one was able to get a permanent residence permit, he might be soon eligible for citizenship (again, this depends on the particular EU country). However, in case of Germany, very often (but not always) one will be needed to cancel all pre-existing citizenships in order to get a German one.
P.S. I'm not a lawyer.