The 310 Cadet's rim is very thin, so an unmodified 32-20 or 32 SWL shouldn't chamber. If the action will close on one, the rifle is probably out of headspace.
The 32 SWL bullets are undersize for the bore, but if the rifling is crisp there may (just) be enough engagement to stabilise the bullet. However, there will be a lot of gas blowing past that undersized bullet, so you should expect lots of leading from gas cutting.
TonyC