Flight Instructor with student reported a near miss with another instructional flight that flew an excessively wide pattern then climbed into the instructor's flight path; requiring evasive action.

Narrative

I was flying in Aircraft X while working for Flight School A ran out of ZZZ. I was doing a lesson in the pattern with one of my students when we had two CESSNA 172s from Flight School B located out of ZZZ1 Also flying in the pattern. They are regulars here multiple times a day. The aircraft were flying extremely wide downwind legs extending out beyond 3 mile left base and final legs. This was making it extremely hard to judge when we were supposed to be turning in our traffic pattern or if they were even remaining in the traffic pattern. This is a common issue that we have with this fight school flying in and out of our airport. We fly typical 45° angle off the back left shoulder traffic patterns and that is how we teach our students here; I understand that other flight schools are different; especially coming from a towered field to an untowered field; but this is just extremely unsafe and unacceptable. On one of our final patterns; we did a touch and go following Aircraft Y in the pattern; we believe that the aircraft was exiting the traffic pattern back to ZZZ1 on the downwind for runway XX based off of his extreme distance from the runway and lack of communication. We began to turn left base once we were at the appropriate position to do so; they immediately after that said they were turning left base also. We then straightened out our course of path back onto the downwind and follow them in. We communicated to them that we were rather close to them and would not be landing and executing a go-around to their right. They did not respond to us and once they were above the runway; they decided to do a touch and go climbing right back up into our path of flight; causing a near miss. We communicated with them immediately and took evasive action by climbing; and turning even further to the right than we already were. The people in the aircraft were extremely disrespectful to our voiced concerns and they departed the area back to ZZZ1.