To shut down the Most Wanted cars you need to get enough SP to move up the Most Wanted List. You earn SP by completing races and smashing other cars.
To unlock the Most Wanted Race Number 6, you need 415,000 SP.
Finish the race in 1st, and then smash into the car after you have finished to unlock the car and the trophy.
Don't worry if you lose the car when you need to take them down, they then appear in the overworld as a black arrow.