The difference between the Krieghoff 32 and Remington 32 is the size of the hinge pins. The Kguns pins are larger. Now some remington 32s have these larger pins as they were upgraded....in fact the two Remington 32s I have both have the larger pins. I beleive this was done by Krieghoff or by gunsmiths later down the line. I don't beleive Remington did this but I dunno. Once I was told that none of the Remington receivers came from the factory that way. You can send your reciever to Krieghoff International in Ottsville or to Norbert at RK Shooting in San Antonio and they can update this for you. Then any Krieghoff Barrel will fit on the gun. Hope that wasn't tooo confusing....
I have shot k-80 and k-32 barrels on my model 32 remington. Only difference is the k-80 and k-32 middle barrel hanger is thicker. You either need to remove wood from the rem model 32 forend or use a k-32 or k-80 forend. I used to swap barrels as a kid(1960's) with my dads rem model 32's and Krieghoff 32 san remo that he had bought new in 1959. The Hinge pins maybe be a problem if they have been changed due to wear. Todd
BTW: Gene knows a lot more about these guns. Maybe he will see this thread.
FYI. I've found the Remingtons to quite good as regards their tolerances. I have four of them and five sets of barrels. The barrels will all interchange on the four actions, but a couple of the combinations are a little too snug (wide). Serial numbers range from something under 2000 to about half way between 5000 and 6000.