I have some Buffalo Bore 340 grn .44mag rounds that I bought for a Dan Wesson I had. I sold the Dan Wesson to a young man at church who was looking for his first bear back-up. I had got the gun for my wife, and she did not like it, too heavy and bulky. Any way, I kept the ammo, and shot 6 of them the other day through my 4 5/8 NMSBH. Kick was stout but managable, accuracy was nice. The question is, the face of the bullets are basically flush with the face of the cylinder. I know that if there is any bullet jump, the cylinder will lock up, but I have a free-wheeling cylinder that turns both ways. Is bullet jump remotely common with this type ammo, am I taking any kind of extreme gamble?

Buffalo Bore is usually very heavily crimped. I shot several boxes through my Super Blackhawk with no problems. I don't think you'll have any problem.

Yea, they load 'em pretty hot so know what it takes to make it work.

It's definitely a consideration as it can happen and when it does, you can be left in a precarious situation. You owe it to yourself to thoroughly test whatever load you choose.