A PA-28 Flight Instructor reported a NMAC in the pattern at a non-towered airport.

Narrative

My student and I were practicing pattern work out of ZZZ running left traffic for Runway XX. On our second take off; I noted that it appeared that traffic was departing Runway XY out of ZZZ1 and instructed my student to extend upwind. I was unable to make visual contact with the aircraft at this time. When the traffic appeared to turn crosswind based on the ADSB data I instructed my student to begin their crosswind turn. In the turn we were able to make visual contact with the aircraft that did appear to be on the left crosswind Runway XY for ZZZ1. We maintained visual contact and began our downwind turn. At this point the aircraft began a 180 turn towards us and passed under our line of sight. Then once we were established on down wind the aircraft made another 180 turn directly behind us on down wind. The aircraft was Aircraft Y. I made attempts to contact the aircraft on ZZZ airport frequency XXX.XX; ZZZ1 air port frequency XXX.XY; the practice area frequency XXX.XZ and XXX.XA. But; I was unable to contact the pilot of the aircraft.