A flight instructor giving instruction in the pattern at a non towered airport reported a NMAC with traffic overtaking them.

Narrative

There were multiple aircraft in the pattern conducting touch and goes for Runway 18 with an inbound light jet. We had the appropriate spacing to line up and wait behind an aircraft which just touched down for a touch and go; with the next aircraft in sequence for landing on a downwind to base turn; once the aircraft in front of us went wheels up we began our take off roll. We had the aircraft in front of us in sight and maintained visual separation; however we noticed on our ADS-B that the aircraft which conducted a touch and go behind us while we were upwind was slowly closing the separation gap until finally when we were well into downwind it appeared they were below us on ADBS-B. I instructed my student to bank right so I could look out the window and the aircraft which had been behind us; which was now below us; began a right turn to depart northeast while accelerating past us. I instructed my student to slow the aircraft until we were behind them and then instructed my student to execute a direct eastbound turn while I maintained visual separation and had my student then perform an extended right 360 to re enter the pattern. We do not have a Mode C transponder or ADS-B out in the aircraft; so we make additional radio calls so other aircraft can create a mental pinpoint of where we are; while we also monitor ADS-B in for enhanced situational awareness.