For an idea how much you COULD switch airframes... I have a friend at United who was a 757/767 FO for several years. In the last 3 years he's been upgraded to 737 capt, downgraded to 787 FO, and just upgraded again to A320 Capt. he also could've bid 777 FO at one point but he knew he'd be bumped off it within a few months and didn't want yet ANOTHER training cycle...
The "airframe assignment" as you put it, like everything else, is based on seniority.
But to even have a chance at any of that, you have to actually get at least one of them to call you and offer an interview. And THAT is by far the hardest part/biggest hurdle.