Having the arrest warrant allows to stop and arrest on its own
The rule for when a stop is allowed is the Terry rule. The police needs a reasonable suspicion for a violation of law to initiate a stop. Having an arrest warrant is better, since that is actual probable cause to arrest: the policemen don't need any current violation of the laws, they have an order to arrest that person at any moment they meet them.
The outstanding arrest warrant alone is enough reasonable suspicion to stop any car registered to the person named therein to see if the driver is that person. That's the fact pattern of People v. Dominguez (1987):
While on patrol on May 5, 1986, he saw a pickup matching the description with a lone male driver. He radioed the license plate number to a dispatcher and requested a warrant check.
The dispatcher notified Johnson that a misdemeanor arrest warrant had been issued on the possible driver of the vehicle. The officer then stopped appellant and asked him for his license. Appellant instead produced a California identification card.
Officer Johnson then requested the name of the person on the arrest warrant which was cross-referenced to the vehicle plate. He was informed the person's name was that of a woman. He was then told a separate arrest warrant, not cross-referenced with the license plate, was out on appellant. Appellant was arrested on that warrant. A search during booking revealed the heroin which was the basis of his conviction.
If the person detained is not the one on the warrant, he or she will normally be briefly questioned regarding the person wanted and the detention ended. Such a procedure is almost identical to being questioned in one's yard or at the doorstep to one's home if contacted there by officers attempting to serve a warrant on a person whose last given address is at that location.
