GA pilot reported a NMAC while in the traffic pattern of a non-towered airport requiring evasive action.

Narrative

Upon initial climbout of runway 30 at Visalia airport; my student; and I; [in] Aircraft X; departing for closed traffic for the purposes of training. This was the second touch and go in the traffic pattern. While in the traffic pattern; Aircraft Y advised that they were 6 miles south west of the field and was inbound for left Closed traffic for runway 30. At this point; my student and I were on a base leg to final for runway 30. After executing our touch and go operation; we began our upwind leg-- at which point Aircraft Z advised of their intention to depart runway 30 and asked us to make an early crosswind turn for their departure. It was at this point that we accommodated the request and announced on common traffic advisory frequency that we were making a left crosswind turn for left closed traffic. Upon making this call Aircraft Y advised of their midfield left down wind entry. While this aircraft did make a radio call stating their intention of entering the pattern; they failed to locate us in the traffic pattern prior to entering their downwind. This resulted in a near miss as we approached our crosswind to downwind turn. Because of our early crosswind; this resulted in a near miss of Aircraft Y; where my student; and I made an evasive maneuver to avoid collision with the aircraft.