Assigning the desktop only seems to work for me for a particular document or session. Or perhaps some applications are not recognizing the setting properly. For instance, if I assign Pages to "This Desktop" whatever that may be, it only seems to apply to that particular document. If I open a Pages document from within Mail, that document will open in the same space as Mail instead of switching to whatever desktop I had previously assigned.
Similarly, if I set Pages to Full Screen, that applies only to that one document, and future documents won't group into a Pages full-screen Space.
Maybe I just don't understand this fully yet, but I'm a bit disoriented.