A flight school instructor conducting training in the pattern at a non towered airport reported another flight school aircraft followed them too closely and had to initiate a go-around.

Narrative

My student and I were at ZZZ alone in the pattern for Runway XX. We were doing a touch and go. Aircraft Y with other instructor pilot (IP) and other student pilot (SP) (private student); entered the pattern and proceeded to do a go-around. Due to their higher altitude and airspeed; allowing them to turn crosswind faster; they caught up to us in the downwind as we turned base. By the second pattern; Aircraft Y was within 0.5-0.75 miles of us according the ADS-B. We landed on the 1000 feet markers and as we touched down; the traffic alert sounded with +400 in yellow illuminating. When I looked behind my shoulder; they were in sight over the approach end of the runway. Aircraft Y called go-around soon after and flew right offset but even on the downwind they were still tailing us closely. I had to ask Aircraft Y to space themselves from us and on the third lap they chose to do a right 360 for spacing. At no point did they extend upwind or downwind in the three laps despite it only being us in the traffic pattern; which culminated in them being within 400 feet vertical of us on landing and them on short final.This is a safety concern due to a private pilot being shown that either he can consistently fly close to other people; or that he wasn't being corrected. Proper spacing in the traffic pattern is crucial for safety. Even if the student wasn't a private pilot; that type of ADM and risk management is unacceptable.