You're in Oklahoma, right, Mark? How local jurisdictional boundaries work varies from state to state and municipality to municipality, from what I recall. Here in urban and suburban Dallas, local cops can pull people over on suspicion of a crime and follow them outside the jurisdiction for the stop if the offense occurred and pursuit initiated within the jurisdictional boundaries. They usually need an on-duty/in-jurisdiction officer to write any citation or do any arrests, just as an off-duty officer can make a stop but needs one who's on the clock to deal with the real administrative details.

I'm thinking you're in the Tulsa area. Call the University of Tulsa College of Law and ask them how that works up there. Or, if you can stand to, call the Sooner School of Law. They'll probably all be out scratching their heads, though.