As far as I understand
- Bans of a user from their instance get federated so that your instance can do any necessary cleanup, as that user will not be able to post anywhere until unbanned, as users are strictly tied to instances.
- Bans and other actions (mods deleting posts) in communities you follow get federated so your instance can take the appropriate action.
- All of this shows up in your server's /modlog, and can be filtered by action and user the action was taken against. It will tell you exactly which mod took which action on which post/user/etc.