Exported everything on my old machine to my NAS, using Calibre’s Export all Calibre data.
Installed extra/calibre. It even reminds you of the import on the welcome screen—nice.
Imported everything from the NAS folder, using the new Calibre’s Import all Calibre data.
It was complaining about a non-empty folder, because I was re-using the new Calibre’s ~/Calibre-Bibliothek folder. Just cleared that out and tried importing again. Worked this time.
Calibre restarted, and all my settings and books are there, like I never changed machines. Nice!