External-drive-housed Footage going offline when switching from desktop to laptop

This has been agonizing: working on a desktop 5k iMac on a project all housed on an external Thunderbolt-connected Lacie, I switch to my laptop to edit from a remote location, and all the media goes offline.  It takes over an hour for the media to reconnect, so I'm effectively hamstrung, on tight deadlines, while first the media all relinks, then further hamstrung while it loads the clips.

What the heck is going on and how to I slay this demon?