One guiding principle in the moderation of SE sites is that users deserve a second chance. This means we avoid shaming them if they were suspended once, in the ideal case the user learns from the suspension and we never mention it again.
I think this is a very reasonable guideline for the common cases, but it does break down for a certain group of repeat offenders. Information about suspensions is not shared across sites, so a user can repeat the same kind of behaviour on multiple sites, and in most cases they will be treated as if this offense was their first one.
In the worst case, this means a user can roam the SE network, misbehave on one site until they're suspended, and then simply move onto the next SE site. There is no mechanism to prevent this, and the moderators on the next site won't necessarily know that the user has behaved in that manner on many sites before.
History matters. We do treat a user that made a single mistake in a very different way than a user that repeats the same offense over and over again. But we don't share that kind of information about users across SE sites. Which means, users misbehaving on multiple sites in the same manner will often be treated just like first-offenders, while they actually repeated the same pattern of misbehaviour again and again on different sites.
To preserve the original idea of rehabilitation, I think sharing the history of user misbehaviour should only happen for repeat offenders. I'm thinking about something like at least three suspensions, maybe even with the restriction that they can't be all on the same site. But from that point on, the guiding principle shouldn't be giving the user the benefit of the doubt, but to provide the SE communities with the information they need to deal with the troublemaker. So a user that meets these criteria would have an annotations visible on their profile (for moderators) detailing all annotations and all suspensions network-wide.
This shouldn't affect users that make a single mistake, but it should mean that users that repeat bad behaviour will face moderators aware of that fact, no matter which SE site they're on.