Flight Instructor reported a NMAC with another aircraft at a non-towered airport while in the traffic pattern; resulting in the flight instructor taking evasive action.

Narrative

I was instructing in Aircraft X with student handling flight controls. We were entering the left downwind for Runway XX at ZZZ. I had a 3rd party (sentry) ADSB-in receiver attached to the aircraft inside the cabin. Upon entry into the left downwind; with radio call announcing our position; I saw on iPad that there was an aircraft on final for XX; I confirmed visually that there was traffic on final for XX. When established in the left downwind for XX; with radio call announcing our position; my iPad announced a traffic warning: 100 feet below; trend heading straight for us. The aircraft; a Vans RV tailwheel; was climbing at a high rate of climb with a left turn that started before the threshold of the runway. I had a visual of this action; took evasive maneuvers to get out of the path of the climbing aircraft; and re-announced my position to the climbing aircraft. The PIC of the other aircraft said 'don't worry we have you in sight we'll be right behind you.' Looking outside; that airplane was </=100 feet vertically and </=150 feet horizontal distance from my 7 o'clock direction. I talked with the aircraft owner on the ground after the flight. Aircraft owner was flying with another pilot rated for the airplane; acting as PIC. Aircraft owner is a student pilot and was not acting as PIC. Aircraft owner was very apologetic of the occurrence; and explained that he was not the one handling controls. The rated PIC was in control of the airplane flying the airplane unsafely. There are previous unofficial local reports of reckless behavior by this certain PIC; of which the student pilot was unaware; that have not been addressed by an enforcing action. Aircraft owner told me he will not be flying with this pilot again; and that he does not recommend anyone else fly with him either.