General aviation instructor pilot reported a near miss with another aircraft when in descent under VFR conditions on a training flight. The instructor executed an emergency descent when visually acquiring the other aircraft to avoid a collision.

Narrative

I was conducting a cross-country flight with my instrument-rated student in Aircraft X; transitioning back to Falcon FFZ after passing over Chandler CHD's airspace. We were flying at 3500 feet MSL when we encountered a Aircraft Y; which was operating under visual flight rules (VFR) and was in the process of requesting an IFR clearance from ZZZ to Falcon FFZ. Aircraft Y had penetrated the Phoenix Bravo airspace (PHX) without clearance and appeared to be having difficulty with communication with Phoenix Approach.When Phoenix Approach confronted Aircraft Y regarding their unauthorized entry into the Bravo airspace; the Aircraft Y pilot did not acknowledge the error; further demonstrating confusion and an apparent lack of situational awareness. At this point; I made the decision to cancel VFR flight following and switch over to Falcon Tower to continue our flight into the airport.As soon as we switched to Falcon Tower; we observed Aircraft Y making an unexpected and rapid turn directly toward our aircraft. At this point; we were in the process of continuing our flight when Aircraft Y started descending into our flight path. We quickly assessed the risk of collision and executed an emergency descent from 3500 feet MSL to 3020 feet MSL; directly over Chandler CHD; to avoid the incoming aircraft.Had we not descended immediately; Aircraft Y would have collided with us without a doubt. Aircraft Y passed directly over our aircraft at a distance of less than 500 feet. Despite the close proximity; the pilot of Aircraft Y made no apparent attempt to avoid the collision. There was no indication from Aircraft Y's flight path that they had us in sight or were aware of our position.Inadequate Communication. The Aircraft Y pilot appeared to be unfamiliar with proper communication protocols; as evidenced by the confusion when trying to communicate with Phoenix Approach. Unauthorized Bravo Airspace Entry. The Aircraft Y had entered the Phoenix Bravo airspace without proper clearance; adding to the overall risk and confusion.Lack of Collision Avoidance. Despite the close proximity of our aircraft and the danger of collision; Aircraft Y did not take any corrective actions to avoid the incident.This incident qualifies as a near-miss collision and highlights a number of serious aviation safety issues; including improper airspace entry; poor communication; and lack of situational awareness on the part of the Aircraft Y pilot. The actions we took to avoid the collision were critical to preventing an accident. However; the event underscores the importance of vigilance and adherence to airspace and communication protocols; as well as the need for all pilots to exercise proper collision avoidance.Given the gravity of the situation; I recommend further investigation into the actions of the Aircraft Y pilot; as well as additional training or review of airspace regulations for pilots operating in the Phoenix Bravo airspace.