The insurance follows the car. Used to work for a major insurer, so I know how this goes, in my state anyway.
In case of an accident, the ONLY way the insurers would not pay up is if an excluded driver was doing the driving. An excluded driver is listed specifically on the policy. Having a person who has a DUI or a bunch of points on their license listed as an excluded driver saves you tons on your insurance.
In my state, every licensed driver who lives in the home must be rated on a vehicle. So, if you have a kid with a lot of points, that kid is rated on the oldest, junkiest, worthless vehicle you have. Oh, and all this isn't done automatically. Watch your policy when a kid turns 16....they will auto rate the kid on the most expensive-to-repair vehicle, until you contact them. They are after a profit, after all.