C172 flight instructor with student reported an NMAC during traffic pattern entry at non-towered airport with a departing aircraft. Instructor took evasive action to avoid a collision.

Narrative

I was conducting a Private Pilot Certificate training flight with my student in preparation for his practical test; scheduled for Day 2. We began the flight on Day 0 around XA30 UTC in Aircraft X. Our transponder was squawking 'VFR;' 1200; ALT verified. Synopsis of incident:The student and I conducted the majority of this flight without incident in the Southwest practice area and at ZZZ1. I was seated in the right-seat; and my student was in the left-seat. We had just practiced unusual attitudes; inadvertent flight into IMC; and he was under the foggles as we returned to the area west of ZZZ and received the AWOS. The student updated both our G5 altimeter and stand-by altimeter with the reported setting. We made the decision to overfly mid-field at 3;600 MSL from west to east; perform a right-tear drop and enter the left-downwind for Runway XX on the 45-degree leg. The student made a 10-mile position report announcing our intentions on ZZZ CTAF. At that point we had completed our descent checklist and before-landing checklist; position lights; strobes; and taxi-and landing lights were activated. He then made a 3-5 mile position report which again; announced our intentions. A Cessna Caravan announced they were holding short of Runway XX. As we were overflying mid-field I observed the Caravan holding short. At that point; I told my student to 'go visual.' I made a position report as we overflew mid-field and again announced our intentions on CTAF. A short distance; (maybe 0.5 miles) from mid-field; the Caravan announced he had 'traffic in the downwind in sight.' There had been no other aircraft making position reports on CTAF that we heard. My student continued heading east at 3;600 MSL; and about 3 miles east of ZZZ began his right descending tear-drop. Not wanting the departing aircraft to be confused and think we were established in the downwind; I made another position report that we were starting our tear-drop and looking for departing traffic. The Caravan announced he was departing Runway 18 and would fly runway heading. I assumed he had heard us and was going to fly well clear of the traffic pattern before heading east. As we reached around 3;300 MSL and were now generally heading west; I heard the Caravan pilot state he was going to make his turn to head toward ZZZ2 and was climbing; and it was his 'last call.' At this point I realized what was about to happen. We were above him and descending; he was below us and climbing; and his turn toward ZZZ2 was going to place him directly into our flight path. I took control of the airplane; and had my student look for traffic; his side would hopefully provide the best vantage point. I made a call that we were leveling off at around 3;300 and observed the departing traffic on our ADS-B in display. The traffic was headed generally northeast; approximately 6-700 feet below us; and climbing. We could not deviate to the right as we were converging; the approaching traffic was not deviating course. A right deviation would have had us continue in the departing traffic's flight path. We continued looking and continued to observe the ADS-B show the aircraft was climbing toward us and directly toward our nose. I knew I was not going to be able to out-climb the Caravan; a climbing turn to the left did not make sense. My student finally observed him at our 10-o'clock and exclaimed 'there he is!' I saw the Caravan pointed at us and climbing; approximately 1-200 feet to our 10-o'clock. It appeared the only way to avoid him was a right descending dive. I immediately pulled the power; pushed the nose down; and banked to the right. A student on the ground advised that he observed the evasive action. I leveled off at approximately 2;900 MSL in the left downwind for Runway XX. We completed the remainder of the flight without incident; promptly landed on Runway XX; and taxied back to the flight school. It should be noted that it was clear; and the morning sun was likely directly in the eyes of the Caravan pilot.