Flight Instructor reported a NMAC while entering the downwind for landing and the need for evasive action to avoid a collision.

Narrative

A cross-country training flight was being conducted from ZZZ1 to ZZZ. I was the instructor. Approximately 14 miles to the north of ZZZ; we reported the field in sight to ZZZ2 Center and were instructed to squawk VFR and change frequency. The plan was to set up on a 45-degree entry to the left downwind of Runway XX. We made regular position updates at 10 miles; 5 miles; and on the 45-entry. Less than a minute after entering the 45-to-the-left downwind of Runway XX; my student and I spotted Aircraft Y taking off from the runway and making a west turn out. We noticed the initial radio call he had made when he was taking off. He reported that he was going to be departing to the west. We were maintaining 1000 ft. MSL on this leg of the pattern and maintained course while keeping an eye on the Aircraft Y that seem to be turning towards us. His initial heading appeared to be one that would cause him to pass behind us by a fairly wide margin. The aircraft continued to turn and when it was about a mile to our ten o'clock low; I noticed that the aircraft was set up on a collision course with ours. My student was controlling the aircraft for almost the entirety of the flight leading up to that point. When I noticed that a collision course existed between the two aircraft; I took control of the airplane and pitched down for an approximate 500 to 1000 ft. per minute descent and leveled off just below 800 ft. Because our aircraft was a high wing; the wings were oriented in a way that did not allow visual contact to be made after the near miss. I estimate that there was about 200 ft. of vertical separation between two aircraft. After it was verified that there was no more collision hazard; I climbed the aircraft back up to 1000 feet and entered the traffic pattern into ZZZ. During the climb back to 1000 ft.; I addressed Aircraft Y over ZZZ Common Traffic Advisory Frequency; notifying them; Aircraft Y that made the left turn out; we had to dive pretty drastically so you wouldn't hit us." There was no response. My student took control of the aircraft and landed without incident.Overall; I would say that our procedures were textbook which allowed us to spot the departing traffic while we were on the 45. Approximately a minute before the incident; I radioed the FBO at ZZZ announcing that I was about to enter the downwind and that I would be parking at the FBO and requested parking instructions. The smaller runway at the airport was closed due to overflow parking and I wanted to avoid any ground traffic conflicts by having a plan to taxi where I needed to. In hindsight; that radio communication could have detracted from my situational awareness in the traffic pattern. I used the opportunity to emphasize to my student that it's not just the people in small planes that are not looking out. Professional pilots can become complacent and cause serious hazards. If situational awareness was compromised and quick action was not taken; there would have been a very high chance of a midair collision."