I did the first recommended 'dungeon', the Rito 'Wind Temple'. It's a floating sky ship, but still Zonai designs. The goal was to unlock a grate in the middle that had 5 locks. As soon as I arrived the map unlocked and there was a cutscene telling me to unlock the locks, and then the 5 locks got marked on the map. You fight your way up to the dungeon like the little events before the divine beasts. You fight up with a companion character who has a special ability (replaces the champion abilities), you use the ability of the character to trigger the maguffins, in this case wind turbines. Once you do the 5 turbines the middle opens and the boss comes out. The boss does not have any unique mechanics and is for all intents a single stage fight with a half health enrage that just adds one more thing to avoid. Completing the 'dungeon' (the game calls them dungeons, but they are not dungeons) unlocks a summon of that companion available whenever, they fight with you and give you access to that characters ability.
There were no actual puzzles, there were a few enemies lying around. They are quite literally divine beats but with different things to collect instead of terminals, and they don't have the 3d map that you control. I don't know how different the other dungeons are, or if there is a fundamental change after finishing the first 4 or what. The only ability I even had to use was the master hand, and that was just to open a door and stick a stick on broken levers. Nintendo may call them dungeons, but they are NOT dungeons, they aren't even shrine puzzles.

Snipped from the complete sky map posted earlier, this is the entire 'dungeon', its 3 floors but the bottom and middle floors are literally a single hallway. Its even simpler than the divine beasts because there is no controlling the map to change areas. There was nothing that was even a puzzle beyond 'stick something on this lever because the handle fell off'. I'm HOPING that this is because this was the first one (game guides you to do this one first) and that the other ones will have actual puzzles or anything. I HOPE that after the first 4, there are REAL dungeons. I am still missing 2 of the abilities (2 missing in the select wheel, but I think there were 4 more slots). But until I get to that part of the game, I am stuck with this, which is a very poor first impression.