Also, the enemy team in multiplayer would have to lose for the campaign to continue
Remember, it's 2 different Campaigns being played at the same time, not just yours.
"If you’re trying to play through a storyline in “proper order” in Brink’s Campaign mode, the game will just work the way you’d expect it. If you fail your mission, you’ll try it again, no matter if you’re playing single player, co-op, or competitive multiplayer. We’re not going to force anyone to play all the missions in order, though. Brink is like a book… when you buy it, we’re not going to try to stop you from turning to the last page and seeing how the story ends. We’ll present it to you in the proper order, but if you want to skip a chapter, or play in any order you like, go right ahead. Our story is written such that you can jump in at any point and understand what you need to know. Lastly, if you’re playing Brink multiplayer in Free Play mode and happen to lose the match, you’ll just move on to the next map in the rotation – much like you’d expect from any other multiplayer game.”
- Richard Ham, Creative Director