Instructor pilot with student reported an NMAC at 06A non-towered airport requiring evasive action to avoid a collision.

Narrative

I was doing pattern work with my commercial student at Moton Field (06A). When initial contact was made [with] Aircraft Y; my student and I were on final getting ready to do a touch and go. Aircraft Y asked us what our intentions were after the touch and go and verified with him that we were going to be remaining left-closed traffic. As we touched down and got back up in the air again; we began our climb to pattern altitude. As we further climbing out on the upwind leg; I noticed that the Aircraft Y was just to our 2 o'clock a few miles out and 1500 feet above us (according to TCAS). The TCAS showed us that he was descending in our general direction and quickly. Over the CTAF I asked the individual in Aircraft Y if he had upwind traffic in sight. He said no; but continued descending towards the upwind leg; with the goal of entering the downwind. At this point I saw that Aircraft Y was on a collision course with us and descending into us. Upon realizing that Aircraft Y was not going to change its course; I told my student that I had controls and took evasive action to avoid a collision. I made a turn to the right and started a descent. As I lifted the wing in the turn; I saw Aircraft Y pass over us around 150 feet above us. Aircraft Y then continued into the downwind leg. The pilot of Aircraft Y did not acknowledge us after that point. Aircraft Y's entry into the pattern was not legal. Per the FAA; entries into the traffic pattern must be over mid-field into the downwind; or; on the 45 into the downwind. Crossing over the upwind leg while there is traffic is not only illegal but is very dangerous. I am certain that the evasive action taken prevented the collision of the two aircraft.