Instructor with trainee reported a NMAC while practicing landings and pattern work at a non-towered airport. The instructor took control and executed a missed approach.

Narrative

A student pilot and myself were on our 2nd leg to ZZZ. We entered the downwind on the 45 for Runway XX over the waypoint; following another Aircraft Y. Aircraft Z (i don't recall the full tail #) on the crosswind passed in front of us following the lead aircraft; we followed the Aircraft Z in a wide right downwind; and announced that we were following Aircraft Z in the downwind. An 'Aircraft A' announced he was crosswind to downwind. I told my student to lower flaps to %50 to follow the Aircraft Z ahead.The first Aircraft Y landed; with Aircraft Z turning base. We extended our downwind to allow for sufficient space to follow the Aircraft Z. The Aircraft A announced their downwind call.Once the Aircraft Z was behind us I instructed my student to turn base and make a radio call. Shortly thereafter the Aircraft A made a base leg call. In my head I thought he must have been closer than I thought; and was turning behind us; while checking the FMS traffic screen. The Aircraft A wasn't present. At this point I was looking outside and to my left for said Aircraft A.I told my student to turn final; and he made the radio call 'Aircraft X turning final'. Shortly thereafter we heard a call from the Aircraft A that he was turning final. Nothing on the FMS or anything visually.Another aircraft spoke up; Aircraft A you just cut a guy off on final; wait he's going around!'. The Aircraft A had just passed into our 1 o'clock low; I called for the controls and climbed and broke out of the pattern to the northwest.My student and I flew northeast to re-enter the downwind on the 45 for an uneventful landing. The Aircraft A made his landing and taxiing calls; seemingly unaware of what had occurred. We landed and taxied back for departure toe the north without incident.