If you have beaten the Wailing Critters dungeon before and your character is at least level 45, Breanni in Legions Dalaran will offer you a breadcrumb quest to fly to the Deadmines in Westfall.
There, Marcus "Bagman" Brown will greet you and send you to the actual dungeon.
The dungeon's 9 fights are much more diverse in comparison to the Wailing Critters, but are overall very comparable in difficulty. If you have a large roster of pets, you won't have any trouble beating it.
The unlocked challenge mode works exactly like before - no healing or reviving pets within the dungeon. Completing it however will grant you one Old Bottle Cap, a new currency that you can directly turn it at Marcus for three new pets:
To get all three (non-tradeable!) pets, you will therefore have to run the dungeon 7 times. Once on normal mode and then 6 challenge modes.
Good luck and enjoy the new dungeon!
Xu-Fu can scan this section for the most suitable strategies according to your preferences and show the required pets.
The Regular option will allow using pets multiple times.
Heroic mode will use each pet only once due to the restriction of no healing.