My ANHA 446 plates originally started on the 89 SE, moved to the Aurora, but my 94 SSEi (AXBF ###) plates and the GTP (BCTR 927) plates were brand new when I got the car. Typically the plates stay with the owner until they are returned for refund for unused time left on the year.
Plates stay with the car here in CA, although I think you have a choice. I've also heard a lot of horror stories of the DMV not switching the info quick enough, and someone racking up a lot of tickets for a car they sold. I think there has been some on here, as well.