The police can run most any plate they want. If they run the plate and find some reason to pull the vehicle over, so be it.
The courts have ruled over and over that there is no expectation of privacy for your license plates, so you take your chance if you drive unregistered, uninsured, unlicensed, wanted, or whatever else. As long as the officer is not trolling for dates and the actionis for a lawful purpose, then your plates and even your name (if the officer knows you) are fair game.