Flight Instructor reported a near miss with another aircraft while on a training flight at a non-towered airport. The instructor maneuvered to avoid a collision; as the other aircraft proceeded to cut in front of another aircraft on final approach after the near miss with the instructor.

Narrative

I'm a flight instructor at a flight school and on the date above; my student and I were doing touch and go's at FHB; an untowered field. As we entered the traffic pattern; we noted 2 small aircraft also doing pattern work at the same time which is not uncommon at this field. We entered the left downwind for Runway 4 and did our first lap without issue and made position calls the whole time. After our first touch and go; we were in the left crosswind for Runway 4 at 1000 AGL and as we were about to to turn downwind; I notice on ADSB that Aircraft X was already in the crosswind and roughly 400' below us and they were cutting us off to the downwind. We called it to their attention and did a right 360 to create space. They claimed that they didn't hear us making calls and couldn't see us on ADSB. Ok; maybe there's something wrong with my ADSB was my initial thought but I stayed behind them in the pattern. While all of this was going on; there was Aircraft Z that had made a 10 mile call and a 5 mile call. I could see all 3 aircraft on my ADSB. I made a call to make it known that I'd be doing another 360 to give space to the approaching Aircraft Z. At this point; we are near the departure end of 4 in the left downwind. Then I hear Aircraft Z make a 2 or 3 mile call and at the same time; Aircraft Y announces they're turning base. So now Aircraft Y has cut off Aircraft Z as well and forced Aircraft Z to fly under Aircraft Y. Now Aircraft Z pilot is yelling at the instructor in Aircraft Y. Again; the instructor in Aircraft Y says that he couldn't see Aircraft Z on ADSB and didn't hear his calls and never saw him. So; my assessment of the situation is that the ADSB of Aircraft Y was not operative but the instructor was not paying attention to the radio or anything else going on in the vicinity of the airport and as a result; caused 2 separate aircraft to take evasive action in the span of about 90 seconds. I feel the actions of the instructor in Aircraft Y were not malicious in intent but they did create a hazard to both myself and the other aircraft involved as well as the student that was with the instructor in Aircraft Y.