Texas Code of Criminal Procedure - CRIM P Art. 45.202. Service of Process

(a) All process issuing out of a municipal court may be served and shall be served when directed by the court, by a peace officer or marshal of the municipality within which it is situated, under the same rules as are provided by law for the service by sheriffs and constables of process issuing out of the justice court, so far as applicable.

(b) The peace officer or marshal may serve all process issuing out of a municipal court anywhere in the county in which the municipality is situated. If the municipality is situated in more than one county, the peace officer or marshal may serve the process throughout those counties.