C172 flight instructor reported an NMAC with another aircraft while in the traffic pattern at a non-towered airport.

Narrative

Began traffic pattern work with my student pilot at a busy ZZZ. As a newer student; I was explaining to him position points at a non-towered field; landing set up and process; and radio calls. Others were in the pattern; one on departure; one allegedly entering onto downwind (P46T) and perhaps one more. We made a call that we were on left downwind for [Runway] XX as we turned onto left downwind. As we continued left downwind; I noted that someone stated that they were also entering left downwind; and looking outside I did not see anyone so I assumed they were behind us. We began to set up for landing abeam the departure end of [Runway] XX and started to descend. We reached 1;800 MSL (800 AGL) and turned final. In the turn; TCAS gave a traffic alert of an aircraft 200 ft. above us. I looked up and to the right as the P46T passed above us on a dogleg entry to final. My student had been at the controls. I took the controls and pitched down to gain some distance between us. This was the evasive action. I descended to about 1;600 MSL (600 AGL) and leveled out and continued to fly to the north. I have a radio call at this point to state that we were departing to the north and would re-enter the pattern. Once I knew we were clear of the pattern I climbed back up to 2;500 MSL (1;500 AGL) to re-enter the pattern in a teardrop formation. As we climbed; I heard the P46T call that they were clear of the runway.