@stephen, sadly I still can't move the windows between spaces. I've tried it with different keyboard shortcuts, and I've tried closing pretty much everything I can close to try ensure it's nothing conflicting, but alas, nothing has worked.
The thing is, it's still moving between spaces, it just doesn't bring windows with it. Any other information I can provide?
If I didn't use my laptop for work I'd just do a fresh wipe and see if it made a difference. I upgraded from Yosemite 10.10.5.