Not claiming right or wrong as I believe the evaluation itself is subjective...
I will say from personal experience that:
HD-650 from my iPhones/iPad airs = terrible. Too soft, and unclear sounding. Might as well pick up a $50 pair of decent headphones for this level of performance.
HD-650 from my FIIO e17 (any player as source) sounds pretty good. I wouldn't want to return them immediately, but there just isn't enough juice to make them loud enough or really bring out the fullness of the sound. Loud enough to hear, but not enveloping as it has the ability to be.
HD-650 from my iFi iDSD Micro (any player as source) really makes the HD-650s sound better than any other headphones I've ever heard. If I never plugged them up to anything else, I'd not feel like I was missing anything. Killer bass, and nice mids. The highs are not painful like in the GRADO SR225e I tried for a while.
HD-650 from my SCHIIT Audio ASGARD2 just puts the icing on the cake. A little wider soundstage, a little more fullness and "weight" at lower volume than the iDSD Micro, and the BASS seems to tighten-up just a bit.
In terms of return on investment as I moved up, I'd say:
50% increase going to FIIO from iPhones/iPad Airs
25% increase going to iDSD Micro from FIIO
And 10% or less going from iDSD Micro to ASGARD2
To set the record straight, neither my Koss PortaPros, UE Super.Fi3s, or my VMODA Bass FREQs sound any better/worse after I go from the iPhone/iPads to the FIIO. The other amps just make them louder, but not noticeably better or more enjoyable. My Photive PH-BTX6 headphones kick up more bass going from the FIIO to the iDSD Micro or ASGARD2, but that's about all besides getting MUCH louder than I would ever listen to.
Just a personal account with no "theory", dogma, or hearsay to clutter things. I do not assert that my findings are universal or empirically prove able /dis provable. Just what I experience. Others are free and expected to feel otherwise.