Flight instructor on training flight with student reported an engine malfunction during cruise and returned to departure airport for a safe landing.

Narrative

I was flying with a student on a cross country enroute to ZZZ1; we had just departed ZZZ. We leveled off at 4;500. Prior to initiating the cruise checklist; we experienced engine roughness. I took controls from the Student and turned back toward the airport. We were about 4 miles west of the field at the time. I tried to adjust the throttle; but it wasn't working. To me; it felt like we vapor locked but I can't be sure. On our way back in to ZZZ; power was restored but the throttle was still unresponsive. I had power idle and couldn't get the plane to slow down. I was indicating around 110 on the way in. Once I knew I could make my landing spot; I cut the mixture to slow the plane down. I executed a power off landing from that point and used my momentum to roll on to the ramp. Tower cleared me to land on XXR as we were coming from the west even though XYs were in use at the time. There were no injuries nor damage of any kind. The aircraft is now undergoing maintenance to determine the cause.