C172 Flight Instructor reported an NMAC while in the traffic pattern on a training flight at a non-towered airport in visual conditions. The instructor visually acquired the other aircraft and took evasive action to avoid a collision. After passing the aircraft; the flight instructor returned to the traffic pattern and landed.

Narrative

I was currently doing an instructional flight with a student pilot; doing takeoffs and landings in the pattern at ZZZ; the home airport for the flight school. We were on the upwind leg from the pattern after doing a touch-and-go and had made a radio call saying; 'Right closed traffic; Runway XX.' Another aircraft - that I am unable to identify and don't know their tail number - reported that they were a few miles to the northeast of the airport and passing to the southwest - did not say altitude. On the upwind leg I could see the aircraft in sight and my student did not. Me and the other aircraft were on a collision course for each other. I told my student we were going to proceed and be ready to turn our crosswind early. This aircraft appeared very low and ended up being at right around 3;100 ft. AGL. Airport elevation is 2;431 ft. I told my student to turn because we were getting very close to the aircraft. The aircraft asked if we had them in sight and I responded with affirmative." My student did not listen to my continuous sayings to turn and I ended up pushing the controls forward and the nose down to avoid the aircraft and we were very close to being directly under them for the passing. After the passing we made a safe return to the normal traffic pattern and landed. This aircraft passed 1 mile to the northwest of the field at 700 ft. AGL with an extreme lack of radio communications. This aircraft later proceeded back to the north doing the same thing at ZZZ1 around 20 minutes later. ZZZ1 is a private airport located 3 miles to the north of ZZZ."