Suspensions are in most cases carried out per-site, users are not banned from the whole network in general. There are network-wide suspensions, but they are not used for misbehaviour of users. Users get a fresh chance on another site even if they have been suspended on another SE site.
So I was surprised to find out that suspensions on an SE site are transferred into chat suspensions, if the suspension happened on the parent site. The parent site can be freely chosen by the user, so it is pretty much random whether the suspension is carried into chat or not.
This suspension transfer doesn't fit to the general suspension policy of SE, and tying it to the rather arbitrary parent user concept doesn't make a lot of sense. I suggest to entirely remove this feature, if a user misbehaves on chat as well, we'll just suspend him there again.
Suspending users from the chat specifically associated with the site seems pretty reasonable, but the problem is that Chat.SE doesn't work that way. If you are suspended from Chat.SE, you cannot chat on rooms for any site except SO and Meta, all SE 2.0 sites use the same chat server.