It shouldn't matter what firearm, or type of firearm, or how oversized the chamber is; a properly made sizing die will re-size the brass to specifications. A small-based sizing die might be required for certain undersized or minimally headspaced chambers, but this has nothing to do with the history of the fired brass case.
Measure a brand new bottle neck case at the base (not rim). Fire it, then resize with normal die. Compare measurements. Have not found one yet that will return the the case to pre fired dia. Course it may depend what you are referring to "specifications".
Yes most sizing dies will size the case enough to rechamber. And after having a go at resizing several batches of 308 brass from a full auto, it most certainly does matter what firearm the cartridge was fired from.