FWIW, I've dealt with this for a long while as well. There are ways to work around it, if you need to keep non-embedded artwork as sidecars, though as 6233638 mentions, it is a bit of a hassle.
It'd be nice if it would treat "simple cases" of Folder.jpg and [Album].jpg as sidecars and moved them like the JRSidecar.xml files. That wouldn't solve all of my cases, but it would solve the majority.
Until such time as the above is implemented, if you need help setting up a scheme where you can move them all together, just ask for help. It takes a little elbow grease to set up at the beginning, but once you get it all set, it isn't bad to handle them via the Rename, Move, and Copy tools. Basically, the deal is you need to import the auxiliary files and tag them so that they "match" in MC (at least as far as is needed to make your renaming rule match). Then, when you move the files, you use a special view where you can see EVERYTHING (including the normally hidden sidecar files) and move them all using the same renaming rule.