General aviation instructor pilot reported a near miss with another aircraft in the traffic pattern. The instructor pilot took the controls and maneuvered away from the other aircraft when they did not follow the traffic sequence instructions from ATC.

Narrative

While making right traffic at TVC for Runway 36 in Aircraft X; I had a near collision with Aircraft Y.Pattern was busy already. I was told to follow aircraft that had extended their upwind. We followed that traffic and made our right crosswind turn. While in the crosswind; the Tower told Aircraft Y to follow the traffic ahead of them in the right crosswind. Aircraft Y confirmed that they had the traffic in sight. Tower then confirmed with Aircraft Y that it was the traffic in the crosswind turn not the traffic in the downwind and Aircraft Y again confirmed that they had me in sight. We made our turn into the right downwind. After rolling out of our turn we got a traffic alert. I looked at the MFD (Multi-function Flight Display) to confirm where this traffic was. At that time I saw that Aircraft Y had turned right crosswind into us and that they were 100 feet above us. I then took control from the student and immediately descended to 500 feet AGL in the right downwind of Runway 36. Then I told the Tower that I was descending. The Tower Controller confirmed that they saw Aircraft Y cut us off. Aircraft Y then continued to make right traffic above us. Aircraft Y asked if they should make a left 360 turn to follow us. Tower told them to continue in the downwind. After this continued for a few seconds I asked tower if I should make a left 360 turn to follow Aircraft Y and Tower told me to continue in the downwind. After that I accelerated to gain distance between myself and Aircraft Y and eventually was able to gain enough distance to comfortably climb back up to a reasonable altitude for where I was at the time. Had I not taken action I believe that this near miss could have very easily been an in air collision.