We have a case on Arqade where a user who is banned from the main site is still chatting in Arqade chat rooms, because his chat parent account was on a different site at the time of the Arqade site ban, and so wasn't affected by the ban. If his chat parent user was Arqade at the time of the ban, he'd have been banned from participation in all of chat for the duration of the site ban.
Instead of working this way, I think it would be nice if users who are banned from a specific site are also banned from participating in chat rooms specifically tied to that same site. So for example, the above user wouldn't be able to chat in Arqade chat rooms, but would be unaffected in other chat rooms.
I think this is cleaner than the current system of banning or not depending on the parent user at the time of the ban.