Escalation will definitely impact the games flavor a lot.
Maybe talking - physical - fighting to injure - fighting to kill.
Or talking - physical - hand weapons - swords. I've played hacks that put the sword in the place of the gun (with the extra d4 and at the top of the escalation ladder). it was the premiere piece of killing technology for a long time, held status connotations, was hideously expensive, and was very effective. That all maps pretty well to guns.
So the hierarchy: Pride, sin, false doctrine, false priesthood, hate and murder. Does this need to change? Are the PCs God's chosen, set to righting wrongs? They better be or Dogs is the wrong game to be playing. The things that need to change are what Pride and Sin mean (see page 61 in my edition at least). The organization of communities may be different, although the principles of stewardship will not be. The sins, and how demons manifest their presence, will also vary a little perhaps, although the list on p.68 looks pretty medieval to me.
You could probably have fun with the doctrine of original sin as well as the complicating nature of the indulgence, which would be a sacrosanct reality throughout Christendom.