Just for one leg it can be like:
At the start of the leg, the teams receive a note telling them that they will mix with other teams, and they have to wait to his new partner (They have a list with the new teams), so each member have to wait till his new partner arrive and make their way.
Each new team will make the leg like if it was with his original partner. They make the RB, the Detour, etc...
Before go to the Pit Stop, they have to stop in some point to wait to his original partner. When both member of the original team arrive to this point, they can make their way to the Pit Stop.
Btw, I like the idea of BritishTarFan, but I don't know if people would like our ideas. xD