Flight Instructor with student reported a UAS hovering over the runway while they were on final approach. The UAS moved as they got closer but stayed within the vicinity of the airport.

Narrative

My student and I were flying a pattern to land runway 18 at MBT. When turning base for 18 I saw a singular flashing red light on the runway over the numbers for 18 and it occurred to me that this was a drone. At this moment the red light or drone was moving laterally across the runway and then just stopped there in the middle of the runway but still near the numbers if not over them. We turned final and the flashing red light started to move down the runway; gaining speed and altitude as it did so and by the time we landed the red light was directly south of the field maybe 1 mile and 500 ft. above the ground. After clearing the runway we came to a stop on the taxi way to do the after landing checklist and I looked south of the field and I could still see the flashing red light directly south of the field and centered on the departure leg for 18 maybe what looked to be 500 ft.-1;000 ft. above the ground and a mile or two south of the runway.I alerted traffic that was taking off and traffic on final for 18 after me. The traffic on final reported the word of what I had seen to Nashville Approach. For this event; I was in Aircraft X. Additionally; the drone did not appear to have any other lights on it.