I'm not an engineering specialist, so I can only make some general comments, though I would like to see some real information show up in this thread. The earcup has a couple of roles to play: first, how it effects the driver, and second and more important, what it does to the backwave coming off the driver. It's not just about space, it's about controlling how the sound bounces around in there. So we're talking housing volume and surfaces of varying reflexivity, but also damping material, foam, and particularly the placement and sizing of openings within that system.
For instance, take a planar driver (like in an ortho or electret headphone), a big and partly reflective open housing, no damping material, and plenty of vents back toward the ear, as in my PMB100 earspeakers: you get an extremely open soundstage with very airy and realistic presentation, but you also get a backwave that cancels a big portion of the bass.
This is an extreme example, built by engineers who wanted to push the envelope in a particular direction (in this case advised by Juerg Jecklin, of Float fame), but in more conventional headphones even very small changes can create big differences in the presentation. Just take a look at some of the blue tak and housing-sealing mods discussed on these forums.
The moral of the story as far as my admittedly basic knowledge can go is that there is no one simple answer to your question: it's not like bigger housing equals bigger bass or something like that. It's just one of the elements that has to work with the others to create the full sound picture. Hopefully others can tell us more.