Campaigns are very similar to single map games but you have a master map containing other enemies and allies.
You select how many enemies and which other players to fight against. Other player's dungeons are controlled by AI and you can upload your own one if you win.
You can press 't' to travel between maps while controlling one of your creatures.
Keepers will find this walkthrough helpful:
When at war with another dungeon, many of the tips in the adventurer guide are helpful for keepers: