C172 Instructor Pilot reported engine problems during cruise. The flight crew requested priority handling and performed an air turn back and precautionary landing at departure airport.

Narrative

We were doing the cross country flight in preparation for my students cross country solo. The plane had just come out of the 100 hour; but everything in start up and the run up looked normal. We took off and made it to ZZZ with out any issues or anything that would cause concern. We landed and taxied back for a right downwind departure of Runway XX; towards ZZZ1. Climbing to 8;500 ft. (which is about 4;300 AGL) everything was fine. We had been in cruise for about 5 or so minutes. At this point my student hadn't done the cruise checklist; and was getting about to the point where I would remind them when we noticed the RPM setting was jumping up; sometimes to 2;900 RPM and sometimes a red X would go through it.At this point I took controls. We were approximately 20 NM northwest of ZZZ; and about 40 NM southwest of ZZZ2. Since at this time I figured it was only a sensor issue and there wasn't any issues with the engine; I made a call to return to ZZZ2. I pulled out my emergency checklist to see if there was anything we could use for this situation; even though I knew there wasn't anything on the checklist for the RPM sensor. We had the pitot heat on already; and as a just to be safe I pulled the alternate static air. I know those or for the pitot static system; and wouldn't effect the RPMs but at this point I just wanted to take every precaution I could.I had not declared an emergency just yet; ATC did ask if I need to declare when I told them we were diverting to ZZZ2; but since I thought it was only a sensor issue I did not. I flew listening to the engine for the RPM settings; but even though the indicator was fluctuation things did not feel any different. The engine was running just fine and there nothing felt wrong with the engine. Also; I had to lean based off sound because when I would try to go to the lean page in the MFD; it would just jump right back to the home screen. No other indications were off or showing any need for concern. Engine instruments all were in the green except the RPMs.Things were fine; until around 15-18 NM southwest of ZZZ2. I had descended from 8;500 ft. to 7;500 ft. due to the Bravo shelf being at 7;800 ft. It wasn't until we were around 7;500 ft. that with full RPM settings we had difficulty maintaining altitude. We were at full throttle; trimmed out; airspeed was indicating around yellow line; and the VSI was around -200 to 0. At this point I [requested priority handling].We flew into ZZZ2; and what was weird was around 6;000 ft. or so the issue seemed to resolve itself. The RPMs were back to normal and all other issues were no longer a factor. So much so when I got on the ground and taxied off the runway; I told Ground we wouldn't need the emergency vehicles and I think they took that as a cancellation of the priority situation.After we shut down; our Maintenance team for the flight school asked me what happened. Then one of them got in the plane with me; we went to the deicing ramp and the maintenance personal set the RPMs to max and they noticed a spike in the RPMs that was abnormal.