You don't 'move' photos from a catalog to another. You have two different catalogs which are totally independent. You can include ("import"), which really means you register and index the photo files into the database. No photo file is moved if you include the same photo in several catalogs. So what you really want to do is to "import = register" a photo in a catalog and then to unregister it from the other catalog(s). The photo file itself does not move at all on your drive.
It's the same idea with albums: you don't move, you add a file to the album (a kind of playlist) or in many albums or you remove it. That does no photo file duplication, and you can remove the file from the playlist of any album without changing anything is the other albums..
So, to keep captions when "importing" the file into another catalog, the captions must be 'written' in the metadata header of the file itself. You use the option of the file menu to 'write metadata' to files, then the caption (or keywords) are imported in the other catalog; same if you "import" the file in another software than Elements.
The analogy of albums and catalogs does not work here with the captions or keywords. Albums don't contain information about the files properties, only a link to the information stored in the database for that file. If you reference a file in a new album, it will access all the properties of that file.