Welcome to Lightroom Community, Matthias. I'll help you with the transition. I'll need more info.
Lightroom & other Adobe apps don't get installed on the external drives & if you copy the installation folder to the external drive, it may not work. Lightroom has to be installed on the internal drive.
What version of Lightroom are you referring to? You're referring to catalogs. Is it Lightroom Classic or Lightroom desktop?
If you're working on Lightroom Classic, check this article for help with moving between computers: https://helpx.adobe.com/lightroom-classic/help/create-catalogs.html#copy_or_move_a_catalog
If you're working on the Lightroom desktop (cloud-based), log in to Lightroom Web (https://lightroom.adobe.com) with your Adobe account & ensure all your image edits & presets are correctly synced to the cloud & when you install Lightroom on the new machine, it should be good to go.
Let me know if this helps,