Pre-Claudius, I believe, centurions could be appointed as
praefecti, to command auxiliary cohorts. Claudius instituted the
tres militiae, which set up a three-tier system of officer appointments for equestrians (the lower aristocracy) and effectively shut the centurionate out of higher command positions (a centurion remained equal - it seems - in rank to a cohort prefect, but there were two parallel streams of promotions).
After Claudius, centurions could still be appointed as
praepositi ('leaders') of irregular auxiliary units called
numeri. But only the rank of primus pilus gave them access to higher commands.
Centurions could be transferred between legions as a promotion, although in the earlier empire this seems to have been arranged by the governors of the provinces and so the transfers only occured between legions in the same province, or in neighbouring provinces. Later (perhaps Nero onwards?) these transfers could be more wide ranging - by the 2nd century centurions were regularly being transferred between legions at the opposite ends of the empire, sometimes repeatedly.
The centurionate, though, was more than just a rank - it was a sort of military caste. The centurionate conferred great privilege, and upon retirement they often took up leading roles in their communities. They were also used for a variety of additional duties: assistants to provincial governors, ambassadors to barbarian tribes, messengers, spies, assassins and more mundane administration roles.
You may be interested in this thesis, which is very detailed but contains much useful information on careers within the centurionate:
Studies in the Legionary Centurionate