The most valuable data (everything you add like markers, descriptions etc., settings) is kept in <user-home>/Library/Kyno, i.e. when moving the tool to another computer, you want to backup and restore that directory. Cache data (thumbnails, metadata), as for any regular mac application, is kept in <user-home>/Library/Caches/Kyno. You don’t need to move it when moving your installation to a different computer but you may. It will speed up a few things.

Also see Does Kyno write the metadata I add into the files?