You will surrender your California Certificate of Title and license plates in exchange for New York Title and plates once you register your vehicle in New York... which you have 10 days to do after moving there. New York will not be collecting expired registration fees on behalf of California. California requires that you submit a change of address form (DMV-14)
within 10 days of moving out-of-state. Unless your move occurs prior to the expiration of the current California registration, the CA DMV will require your pay registration renewal fees. So the short answer is, you can register in NY with the expired CA tags, but CA may choose to demand past registration long after you moving out of state if at anytime while you were in the state your registration was not current.