Currently, moderators have the ability to reset a user's username and/or profile text, and propagate that to the rest of the network, resetting it everywhere. However, often, the user's stackexchange.com network profile will also need to be reset, and this is currently only possible to do by either the user in question or staff.
Instead of bothering staff every time, moderators should have something available similar to chat, where moderators can change a user's parent site and have access to a "refresh profile" button, which will fetch updated information such as username and avatar from the parent site.
Currently, the only thing that moderators can do on stackexchange.com is to see hidden accounts. It would make sense for moderators to be able to refresh profiles there, like chat, and avoid pinging CMs every time this comes up.