Currently, the Media Management page does not consider a clip that is solely referenced in a Fusion comp and not present on the timeline as "Used Media". So when you copy Used Media and import it into a project on another machine, you end up with a Media Offline in every Fusion comp that is combining two pieces of footage.
There should be a toggle on the Media Management page where I can choose to "include media referenced in Fusion" when exporting, so that such clips will be copied, too. Or, if that's too hard, at least in the Offlined mediaIn nodes, their ClipNames should not be blank: they should tell me the name of what clip is supposed to be there. Presently, a mediaIn that has been offlined in this way has a blank ClipName, which makes manually reconnecting the footage effectively impossible.
Please see this forum post for more details:
viewtopic.php?f=21&t=223231&p=1155420#p1155284