Return to Karazhan is being conceived as something like a 5-player raid, and as its difficulty (not to mention time commitment) are higher than the existing Mythic dungeons, Blizzard has said that rewards will be appropriately high as well.
Right now, it looks as if the first half or so of the bosses within will drop 855 gear, which is 5 levels higher than Emerald Nightmare's normal mode. Later bosses will go up by 5 points to 860.
With nine bosses, Return to Karazhan is a dungeon that Blizzard says they expect players will run in chunks, much like a raid. This is really the first time since vanilla that there has been a dungeon designed to be completed in multiple sessions (actually, not even then - they just expected you to slog through the dungeons for hours upon hours. Sunken Temple used to be about three times as big, for example, and took a very long time - not to mention that just getting into a lot of vanilla instances meant traversing an elite-filled area outside the instance proper.)
With that in mind, though, I really wonder how Karazhan is going to fit into the current Mythic+ dynamic. I'll admit I've only run one such dungeon, and due in large part to the fact that it was the second time I had run Court of Stars, we didn't make it within the time limit.
But with a dungeon where standard runs are expected to last multiple hours, I really have to wonder how Mythic + is going to work for it.
There is also, of course, the fact that the gear rewards for this dungeon are going to be inherently better than other Mythic dungeons. When introducing the Mythic+ feature, Blizzard explained that this would allow them to introduce new dungeons at the same difficulty level as existing ones, and simply allow them to be thrown into the mix with the 7.0 dungeons. Return to Karazhan does not seem to follow that model.
On the other hand, perhaps they're treating this as a special case. This is, after all, a dungeon that is headlining a patch that also contains a raid, and seems to be on the experimental side, even if it's recreating one of WoW's most popular raids as a dungeon (and again, is leaving the raid intact - something I hope they continue to do with any revamps going forward - much as a I love the new Scholomance, I do miss the broader and more labyrinthine original.)
In terms of progression, I think that raiders who are satisfied with what they have out of Emerald Nightmare are most likely to head on to Trial of Valor, though splitting the raid to run Karazhan might not be a terrible idea, as I imagine you'll get more gear out of a 9-boss instance than a 3-boss one. Still, we do not yet know what the loot quality of Trial of Valor will be.
All of this is, of course, also subject to change. I know my guild hasn't even started running regular mythic dungeons yet (something I'm hoping we can start doing soon) let alone going into Emerald Nightmare on Normal mode, so I'm in no rush to get these new instances just yet.