Small aircraft Flight Instructor reported a near miss with another aircraft in the traffic pattern.

Narrative

My student an I were in the pattern practicing landings well before Aircraft Y called up on CTAF. When Aircraft Y called up; it was unclear about what kind of an entry they were going to make; and where exactly they were coming in from. The radio call was poor quality as well as their communication. I had them on the ADSB In the whole time so I was not worried about it. After the touch and go in the up wind; I started keeping my eyes out for that same Aircraft Y. Turning Crosswind to downwind; I noticed the Aircraft Y pilot not climbing above pattern; altitude; and heading straight for downwind from the opposite side to maneuver straight into down wind. The Aircraft Y pilot obviously did not hear me and my student making radio calls in each leg of the pattern and failed to realize we had the priority. The pilot overflies the field 1000 feet of a field elevation At our same altitude. As the pilot begins; making his left turn into downwind he makes now is his second radio call at the field. Me having my eyes on him the whole time on my GPS and visual; I made the determination that there was an eminent risk of mid air collision. In the downwind I take controls from my student and make a sharp right 360 avoiding collision with the other pilot. after the 360 me and my student rejoined the downwind from which we continued to a normal lap in a pattern without incident. The separation between the two aircraft must have been between 300 and 500 feet horizontal.