I'm trying to install a heat exhaust vent on the side of the drywall in the basement, see pic. The issue is that the the register is too small for the boot. This is normal if the exhaust is in the floor but not so great on the side of the wall. The question is how to attach the boot to the register to avoid air leaks... Basically to do it right.
You should be able to find a piece of duct at the big box store to extend the boot to the wall. You'll need to trim it with a good set of tin snips. When you trim it leave some tabs so the register screws have something to bite into instead of just drywall. Get a larger register that better matches up to the duct. If you can't get the duct to line up with the register then see if you can rotate the boot and cut the duct at an angle. If you don't do a great job then get some foil duct tape to cover any gaps.