DA20 Flight Instructor on training flight with student reported an NMAC with another aircraft on approach at a non-towered airport.

Narrative

Aircraft X failed to maintain adequate separation in the traffic pattern; specifically on final at ZZZ. I am a CFI and was conducting a flight lesson with my student in Aircraft X; a Diamond DA20. After reviewing maneuvers in our practice area for the flight lesson; my student appropriately established us on a 6-mile 45-degree entry for the right downwind of RW XX at ZZZ. He promptly made a radio call once established on the 45; specifying he was 5 miles out. Shortly after; I heard two other aircraft calling and reporting a 45 entry. Looking down at my ADS-B display; I observed that both aircraft were behind us. I responded with another radio call; updating our distance of 3 miles on the 45. Shortly after; we joined the downwind and made a radio call. One of the aircraft behind us; a Cessna; appropriately reported doing a 360 on the 45-degree entry to maintain spacing. However; the other aircraft; Aircraft Y; a Diamond DA42 Twin Star; reported established downwind. Since there was another aircraft on final; my student slightly extended his downwind before initiating and calling his base turn. At this point; I noticed that Aircraft Y was almost directly above of us on my ADS-B display; but still at pattern altitude while we had descended as we're about to turn base. We then turned and made our radio call for final as my student correctly configured our aircraft for landing and reduced our speed to the appropriate landing speed. Shortly after; we heard Aircraft Y make his final call; stating that he was 'number 2.' Just as we touched down; I heard Aircraft Y calling a go-around and 'breaking to the right'. I happened to glance at my ADS-B display as I heard that radio call and saw Aircraft Y was again almost directly above us before initiating a climbing right turn towards the downwind. While my student completed his touch-and-go and established himself in the departure leg; another pilot remarked; 'That was close.' Aircraft Y responded; stating; 'That's what happens when you extend the downwind 3 miles.' We then continued through another lap or so in the pattern. I realized that we were being closely followed by Aircraft Y; which was uncomfortable to say the least; but I had not fully comprehended the gravity of the situation until I conversed with some of the ground personnel and other flight instructors that were witnessing this take place.After we landed; one of our flight school's ground personnel described to me from his perspective; how close Aircraft Y was to us before they executed a go-around. Another flight instructor and student who were taxiing back to the active runway at the time also described to me that Aircraft Y was only a few feet behind us. Both were shocked at what was taking place. In my opinion; Aircraft Y failed to maintain adequate separation from us within the traffic pattern. Despite having numerous options to ensure a safe distance; they persisted in following us all the way to the runway; allowing the spacing to become unsafe. Given their proximity to us; they had a limited margin of error; and any minor error could have resulted in a catastrophic outcome.