A Flight Instructor reported a NMAC entering the traffic pattern at a non towered airport.

Narrative

Approaching ZZZ from the south a planned arrival crossing from east to west at 5;800 ft; 200 ft. above pattern altitude for Runway XX and 200 ft. below the Class B at 6;000 ft. above. Stated these intentions 3 different times on CTAF. The last announced crossing mid field to head west and reenter pattern on a 45 to downwind to enter for a right pattern to [Runway] XX landing. We planned to cross over the pattern which is 5;600 ft. One other aircraft was in the pattern coming from south to north on downwind. His call was downwind and departing the pattern to the northwest. Unfortunately he continued on the downwind leg to the north and climbed while in the pattern to 5;850 ft.My student in the left seat saw the plane and climbed to 6;000 ft. and we missed each other but very close. I was able to meet with the other instructor and discussed climbing above pattern altitude while on downwind when an aircraft is crossing mid field. It was clearly obvious he understood the danger of what he allowed his student to do. Pilot Handbook of Aeronautical Knowledge (PHAK) mid field crossing and stating your intentions is the preferred entry. Climbing above the pattern to depart the area is not acceptable. Fortunately no one was injured.