Ohio Revised Code Title XXIX. Crimes Procedure § 2935.02

If an accused person flees from justice, or is not found in the county where a warrant for his arrest was issued, the officer holding the same may pursue and arrest him in any county in this state, and convey him before the magistrate or court of the county having cognizance of the case.

If such warrant directs the removal of the accused to the county in which the offense was committed, the officer holding the warrant shall deliver the accused to a court or magistrate of such county.

The necessary expense of such removal and reasonable compensation for his time and trouble, shall be paid to such officer out of the treasury of such county, upon the allowance and order of the county auditor.