Light aircraft instructor pilot reported an NMAC with another aircraft in the pattern at GZH airport.

Narrative

Instrument training cross country flight from ZZZ to GZH. Cancelled IFR at beginning of RNAV 19 when field was in sight. We saw on ADS-B that there was an aircraft in the pattern landing runway 28 so we decided to continue VFR and overfly the field then execute a right teardrop to enter a left downwind for runway 28. Aircraft in the pattern were not making frequent radio calls and we realized around 8 miles from the field that there were 3 aircraft in the pattern. Overflew the airport; flew southbound 4 miles and executed the right teardop to enter the 45 for runway 28. We were very careful to make very clear position reports; attempting to work with the aircraft in the pattern to work in. Two miles south of the field about to turn into the downwind; one of the aircraft turned left crosswind toward us. We immediately turned into the left downwind about 2 miles from the field and made a position report and advised the other aircraft that we had them in sight. Aircraft flew directly above us and cleared us by 200 feet according to ADS-B. We departed the pattern southbound after extending our downwind roughly 3 miles.