Flight Instructor reported while entering the VFR traffic pattern encountering a departing aircraft turning into them resulting in both aircraft taking evasive action to avoid a collision.

Narrative

CPLME (Commercial Pilot Multi-Engine) check flight; SP (student pilot) was at the controls during VFR gap arrival. ATC instructed us to make left closed traffic runway 21. Aircraft Y was cleared for takeoff for left downwind departure runway 21 around the same time. They were not advised of us coming in from the gap despite us on a flight path to intersect their flight path. We were also not advised of traffic departing but IP (Flight Instructor) heard Aircraft Y get cleared for takeoff and was looking out for traffic. While about to make a right turn to enter left downwind runway 21; we received a traffic alert with no altitude readout; less than 1 mile at 12 o'clock. The traffic was on a climbing left crosswind as we were joining at the 45 degree entry. Just as we spotted the traffic; the traffic started a descent about 100-200' below us and deployed his aerobatic smoke; likely in an attempt to get us to spot them. Total slant distance between our aircraft and the target aircraft was around 500' or less. No traffic alert from ATC until after we had both taken evasive action in the form of turns and descents. ATC then asked Aircraft Y to 'say altitude' and then advised Aircraft Y that they were not getting an altitude readout from them. ATC seemed to not think that their departure and our arrival would cause a traffic conflict but was incorrect.