I've only tried a couple of games--Dirt 3 and The Witcher 2--and both are basically unplayable due to a disgusting amount of input lag. I never played these games on Xbox 360, so is this just what they were like? Is it an emulation issue with these games specifically? Or is it a more widespread issue?
Oddly, it seems results vary from person to person. I wonder if hardware variations play a part.
Have you tried resetting your XB1 then playing? It has been reported that resetting (i.e. clears system cache) can make a real difference in performance in some games.