A Flight Instructor on downwind leg to a non-towered airport reported a NMAC with another aircraft on downwind which took evasive action.

Narrative

After completing a practice approach to runway XX in ZZZ with a plan to circle XY another airplane became a conflict. The wind was favoring runway XY. The approach was broken off about 4nm south of the airport to enter the left downwind as there was an airplane departing runway XY. As we entered the left downwind the aircraft on departure made a radio call that they had us in sight and were going to follow us in the left downwind for runway XY. This aircraft was a Cirrus. Our altitude was at the circling altitude for the approach; which is 1360 msl which is 737 agl. In the downwind leg we received a traffic alert from the G1000. The MFD indicated traffic at our 6 o'clock position at the same altitude. We immediately began looking for the traffic but did not see them. A few seconds later I noticed the Cirrus executing a climbing right turn I presume to avoid hitting us. They joined a wide left downwind. The aircraft was within a couple hundred feet of our position when I saw the Cirrus performing this maneuver. We continued the circling maneuver and on short final executed a go-around; which was the briefed plan for the training exercise. The Cirrus joined a wide left base and continued the pattern after we performed the go-around. Our flight continued without any further conflict.