Round specific: They are dedicated to one specific round so some dont like that. Different round, your custom dial will not be accurate with it. Though as some point out, you can try it with a similarly ballistic round but I would personally not do so when hunting.
Elevation: Changes in elevation will effect your ballistics and you have to set your custom dial based on a specific elevation (actually a range in elevation). How much elevation affects your ballistics?, well, I am sure you can start a bar fight over that question but I think most would say you are good within a few thousand feet range of where you set it initially. My problem would be living at sea level and using a dial for around here, it will not be the right set up for 10000 feet in the Rockies, or Tajikistan or wherever.
I have tried a few CDS and I really want to like them but I always switch back to traditional dial with a range card. I also like that it forces me to understand my ballistics better when using a range card, etc... I really like this when teaching my son with his rifle. I cannot imagine a scenario where if the CDS goes down one is lost on how to shoot accurately.
Cheers