SR20 Flight Instructor reported being cut off during the turn to downwind by a trailing aircraft that turned crosswind in front of the Instructor and Student. The Instructor took evasive action to avoid a collision.

Narrative

A near mid air collision was avoided by a small margin. My student and I took off of Runway XXL at ZZZ and complied with noise abatement procedures and stayed in right traffic North of the field since the Tower was closed. There was a second aircraft that took off behind us (a Piper Archer) and a third one (a Cessna 180). We began our cross wind turn at 2;000 ft. as we continued on our crosswind leg; it was time to begin to turn to the downwind and subsequently level off. At the same time I looked at my ADS-B in data displayed on my iPad in front of me and it was blinking in red. The Cessna turned crosswind inside of our pattern and it was headed directly towards us; while the Archer who was number 2 behind us was following us. There is no way for my student to have seen the traffic since they were sitting on the left seat. I took controls from my student and immediately began a rapid descent. The collision was avoided by approximately 150 ft. I called the aircraft on the CTAF and they did not respond. I took the landing from my student and the flight continued without further consequences.