RV-12 Flight Instructor conducting a first flight with a student reported due to a loss of engine power they diverted to the nearest suitable airfield and landed.

Narrative

I was conducting a flight instruction flight with [a student]. It was [students] first flight and this was considered a discovery flight. We had gone around the west side of ZZZ1 past ZZZ2 and were headed back to ZZZ3. [Student] was flying at this time as I was trying to let him fly as much he could handle it. The engine got a little noisier and then seemed quite rough. I immediately took controls and began to fly the aircraft. I was checking the engine instruments and noticed the RPM was dropping but everything else looked fine. At first we only lost a couple hundred RPM. After a few minutes we began losing more and more RPM. This first started over the ZZZ4 near the center of the island of ZZZ1. At this point I turned to set up for a base at ZZZ which was not too far ahead. I immediately contacted ZZZ Tower and told them I would be landing on their airfield. This was a military airfield. They cleared me to land at first. I was slowly losing altitude and more and more RPM as I approached ZZZ. When I turned final at ZZZ I got a little boost in RPM. At this time ZZZ also said that I could not land at a military airfield unless it was a priority and there would be lots of paperwork. I at that point [requested priority handling] and said I would try to maintain altitude to see if it was getting better. However; the engine sputtered again and I told ZZZ that I would be landing. I was a little high so I circled at the approach end and landed the plane. I believe I did the right thing in this situation. Something that I could have improved on was maintaining best; glide speed. However; the engine still had some power. I also probably should have put the plane down and not tried to delay the inevitable when I got the boost of RPM. Even if the plane began to fly normally after that; I would still be landing at ZZZ because of the previous occurrences. I also think that being at 2500 ft. Helped me in this case. If I were even at 2000 ft. I might not have made the runway at ZZZ. Also; I was previously in the military so the military airfield was not that intimidating to me. However; the way ZZZ mentioned on short final that there would be lots of paperwork may have scared some people. So I like to tell people to not be intimidated by military airfields and that an airfield should always be the first option in engine failure.