General aviation instructor reported a near miss with another aircraft; who was transitioning the airspace at traffic pattern altitude; at a non-towered airport. The instructor took the controls from the student and maneuvered away from the other aircraft.

Narrative

I was with a student; and we were departing runway 26 at BHC. This was our 3rd pass in the traffic pattern; and my student and I had been making consistent traffic calls. As we were about to make a left crosswind turn to stay left closed traffic for 26; I was alerted on my EFB that there was traffic at the same altitude; very close. I took the flight controls and dove down to get out of the way. As I dipped my right wing; I was able to see a plane in the same path as we were; at the altitude we were at before evasive action. We re-established into the traffic pattern; and I asked the aircraft; Aircraft Y if they were on frequency. They responded that they were; and they had us in sight the whole time. However; they had not made any calls. They said they heard us making left crosswind and they were going to stay out of our way; they were just transitioning the airspace. The problem is; they were transitioning in airspace at traffic pattern altitude and not communicating their intentions. Even if they saw us; they got way too close to us. I am not certain they really did know we were there; as their flight track seems to indicate they didn't take any action until it could have been too late.