When a university appoints a faculty applicant, the rank is based on his/her current position or the university will judge about the academic rank independently?

I understand that each university has its own regulations for academic rank/promotion, but how much they consider the current position of an applicant?

This should be a serious issue when an applicant comes from another country with a different academic system.

Is it (even rarely) possible to appoint a currently full professor as assistant professor or vice versa?