Sheriffs, under-sheriffs, and deputy sheriffs, may execute all process in their hands at the expiration of the term for which such sheriffs were elected, the execution of which having been begun by him, and shall make due returns thereof in their own name. In case of a vacancy in the office of sheriff, every deputy in office under him may execute any writ or process in his hands or in the hands of such sheriff, at the time such vacancy happened, and has the same authority, and is under the same obligation to serve and execute and return the same, as if such sheriff had continued in office.