Mich. Comp. Laws § 801.110

Sec. 10. If any prisoner confined on civil process, shall have been admitted to the liberties of the jail of the county for which such designation shall have been made, previous to such designation, they shall, notwithstanding, be entitled to remain within such liberties, but may be removed to the jail so designated, and confined therein, by the sheriff of the county in which they were admitted to the liberties of the jail, in the same cases, and in the same manner as such sheriff might by law confine them in the jail of his own county.