N.Y. Jud. Law § 791
The clerk must immediately issue a warrant, under the seal of the court, directed to the sheriff of the county, and commanding him to collect from each of the persons named in the schedule annexed to the warrant, the sum therein set opposite that person's name; and to pay over the sum collected to the treasurer of the county. The clerk must include in the said annexed schedule the name of each person who has been fined, prior to the issuing thereof, and whose fine remains then wholly or partly unpaid, and not remitted by the court. The warrant is the process of the court, by which the fines were imposed. If a delinquent resides in another county, a separate warrant, for the collection of the fine imposed upon him, with an appropriate schedule annexed thereto, must be issued, in like manner, to the sheriff of the county where he resides.
N.Y. Jud. Law § 791