In a county placed in a different or additional judicial district by reapportionment under this subchapter, the district clerk, sheriff, constables, county attorney, and district attorney or criminal district attorney of the county, and their assistants and successors in office, shall be the respective officers of all district courts of the county, including the courts of the different or additional judicial districts. Each officer shall perform all the duties and functions of his office relative to all the district courts of the county.