I think there are doctrinal differences going from D-R to KJV, I’m not so sure about going from KJV to D-R.
One big difference is that the KJV was based upon the Stephenus Greek text from the 16th Century for the NT and the ben Asher Masoretic Hebrew (from the 10th Century). For stylistic purposes, it also copied a lot from the Douay.
The Douay Rheims, of course, was based on the Vulgate.
Interestingly, though, there would have been more ancient Greek and Hebrew texts available to Jerome than would have been available to Erasmus or Estienne.