General aviation Flight Instructor with student reported a near miss with another aircraft in the traffic pattern at a non-towered airport. The instructor took control and maneuvered to avoid a collision; then completed a traffic pattern to a safe landing.

Narrative

My student and I entered an 8 mile 45 for the left downwind of the landing runway at AUO and we were making standard radio calls at 8 and 3 miles as well as traffic pattern calls. As we entered the left downwind I noticed an aircraft crossing mid field about 500 feet about us and heard the aircraft make a radio call. I didn't visually see this aircraft but I could see them on my ADS-B in. About the same time I saw them on ADS-B; a Aircraft Z reported that it was departing below me and that it had me in site so I immediately started looking for that traffic. Seconds later; the Aircraft Y crossing mid field made a call that he was turning left downwind from mid field. At this time I got a traffic alert and saw that his ADS-B track was directly over the top of us and reading 100 feet above us. I asked if he had us in site on the CTAF frequency to which he replied 'No.' While this was happening I took the controls from my student and started a rapid descent to avoid the overhead traffic but quickly realized that I still had to the departing Aircraft Z below me. My student and I flew the rest of the pattern about 200 feet lower than usual and did not see the Aircraft Y until he landed in sequence after me.