Team shuffling was a admin command on servers. Usually, either when the teams have had the same players for awhile, or teams were uneven in either actual numbers or skill, the admins would be able to shuffle the teams, which just placed everyone randomly and evened the teams out. A lot of times they would call a vote before hand, to see if it was really wanted/needed. It's a mechanic I always thought was cool and should have made it's way to console by now.
Oh, that... no, they never made that on console.
Some games have allowed for team swapping (BF2:MC for example) while others you can hack a team swap (BFBC2, at least, you can "join session" using the Xbox menu of a player on the other team)
Some games also had a vote to kick system (BF2:MC again) though that often got abused by clans who would use it to take over servers as their own.
What you describe wouldn't work for Brink MP in campaign mode though.... I thought you were referring to a system that shuffles players across multiple servers, keeping them on the same team, and placing them into a game according to their mission progress, availability, and skill
Oh, on console there IS autobalancing though. Halo, and CoD ustilize it. Between matches it splits the players up based on recent points/kills per match. It keeps friends together, so if the 3 best players are in 1 locked "squad" then the next 5 best will be on the other team