I like both my HD600s and NS1000s a lot. The HD600s are more capable socially, but need a decent amp to push past the NS1000s.
Although the NS1000s are quite fussy about which amps drive them well in passive mode, I was very fortunate to stumble across trying them with the FiiO E5. If you wanted a combination to demonstrate what a huge difference an amp can make to headphones, this would be good candidate. Added to that, the ANR (effectively using a built in amp) mode sounds pretty good out of most sources, although thanks to the 300 ohm input impedance, they can lose detail if driven by something that lacks good voltage swing. The hiss in ANR mode can be a pain if you listen to anything with quiet passages.
The HD600s certainly don't sound their best out of an e5. They sound alright out of most sources and amps, certainly not the bloated mess that is inadequately driven passive NS1000s. I currently have my HD600s plugged into a Mackie mixer, which at least manages to drive them cleanly. My e5, cmoy, SU-DH1, bravo tube amp and Sony AV amp all failed to do that (although I must admit I didn't notice the distortions at first). I was going to get a higher spec amp to go with the HD600s but then thought to try the mixer I use for guitar monitoring and was pleasantly surprised. By comparison the NS1000s in passive mode sound dull and slow out of it (same with the Bravo although both are good for bringing out detail on the NS1000s in active mode). Obviously the two headphones have rather different amplification requirements. It's why I advocate matching amps to particular headphones. It's not just down to sound-signature synergy.
So... I'm interested to hear how the DG does with passive NS1000s. It might be great or might fail miserably.