The Magnagate dungeons are different to normal ones in that you don't control any of your Pokémon when in them. Instead, you get to control any of the five starter Pokémon or their evolutions. These are chosen automatically with any number of partners based on those same Pokémon. Your task is then to go through the dungeon.
There are two different mission types, 'Magnagate Found!' or 'Explore the Infinite Dungeon' but they don't have much different. As you reach a certain floor, based on the length of the mission, you will face up against a few Pokémon. Once they are defeated, you can choose to end the mission or to continue on through the dungeon indefinitely.
You cannot recruit Pokémon found within the dungeon. However, all items and money obtained in the dungeon will be placed in your chest for access within the game. This is especially handy as there are many items found in these dungeons that don't appear as easily in the main game.