Flight Instructor conducting a flight review reported an NMAC in the traffic pattern with an aircraft that was transiting the airport at traffic pattern altitude and was not communicating on frequency.

Narrative

While conducting a flight review in VFR conditions; we departed Runway 30 at SMQ to remain in the traffic pattern. After making a CTAF call; we entered the left downwind for Runway 30. An ADS-B alert on my iPad alerted to traffic at our same altitude off our right wing just as we rolled out in the downwind. I immediately took the controls to raise the right wing only to see another small aircraft intercepting our flight path at our altitude. I immediately made the decision to make a steep left hand turn to avoid a potential collision. After the turn to avoid was made it was noted that the same aircraft made a turn to follow; so we made another steep turn to evade the aircraft.This aircraft was not entering the traffic pattern but rather crossing mid field of the airport at pattern altitude. ADS-B shows this aircraft proceeding to a different airport 20 miles north. The other aircraft gave no indication that they had seen our aircraft at any time during the incident nor were they communicating on the CTAF frequency.