C152 Flight Instructor with student reported multiple and frequent drops in engine RPM during a training flight. Flight Instructor diverted and landed safely.

Narrative

This flight was a standard training flight for a student working on their instrument rating for ASEL at a part 141 school. This student had just passed the phase 4 progress check and was going to do their first flight flying multiple practice approaches during the flight. Prior to the flight I requested the FBO add 10 gallons of fuel per side. During preflight I sumped the fuel and visually verified that the fuel was free of water/contaminants. We conducted a standard run-up bringing the power to 1800 rpm and noted that the mags had a consistent drop between the two of them; the left mag dropped from 1810 to 1760 and the right mag dropped from 1810 to 1750. Prior to takeoff I had Person A run a 'floor to door' check. This consisted of checking in order the fuel selector valve; fuel shut off; trim; flaps; mixture; lights; mags; doors windows and seat belts. We climbed to 3500 msl; leveled off; set power to 2350; flew southwest bound and requested the LOC XXR ZZZ practice approach. I noted at this time that Person A had not leaned out the mixture between 3000' and 3500' in the climb. Received vectors to fly a heading of 180. I heard an RPM drop which lasted less than one second and was around 50 rpm drop. After around 30 seconds I heard another RPM drop visually confirmed the RPM drop as I was watching the tachometer. I asked my student if he noticed this and he responded that this was the third drop he had noticed and if I had messed with anything. I responded that I had not changed anything. I visually verified that the fuel selector was in the 'both' positions; the mixture was rich; and the power lever was set. After the next drop of around 50 rpm which occurred approximately 15 seconds after the previous drop; I instructed Person A to remove the view limiting device he was wearing and put the electric fuel pump to the on position. I instructed Person A to make a left turn to a heading of 030 to point us towards ZZZ1. I made a call to ZZZ approach and canceled the approach request and stated our intentions to return to ZZZ1 . Control asked us to make a right turn and then to proceed on course to ZZZ1. In the right turn the engine made several more drops of 50 RPM. I called approach control and preemptively [advised ATC]due to the nature of the flight being night time; over the city; and lower altitude. After [advising ATC] and while Person A was flying I began loading the ILS XXR into ZZZ1 as a back up to the visual approach. While I was loading the approach the engine dropped from around 2380 RPM to 2200 RPM and remained there for several seconds without a power lever change. I took the flight controls at this point and told ZZZ approach that we would like to divert to ZZZ and land on runway XXR. ZZZ gave us a heading to point us towards the field and I had Person A run the before landing checklist. I kept power set and maintained an altitude of 3500 feet MSL until we were about 3 miles to the south west of the field and I was confident that we could make the runway with no power. At this point I reduced power to idle and conducted a normal approach to land. After landing the engine did not show any signs of roughness at a lower RPM. I believe that [advising ATC]was the correct course of action due to the additional risk factors that this flight had being night time over a populated area and at a low altitude. I believe that diverting was also the correct choice because I wasn't certain that the aircraft was going to make it back to ZZZ1 as it would have increased our flight time by several more minutes. If I encounter engine roughness again I will check the EGT and CHT to see if there is any indications of a cylinder failure. I did not attempt to trouble shoot the engine roughness beyond what I did as we were preparing for a total engine loss within the span of 1-2 minutes. Had I had more time and altitude; I would have run checklists for a rough running engine.