A Flight Instructor descending on an RNAV Approach reported a NMAC with VFR traffic overtaking them.

Narrative

While on approach to ZZZ RNAV XX (filed IFR for training approaches); nearing the final approach fix; my student and I observed an ADS-B contact who appeared to be overtaking and intersecting our path from below and from our left. I (right seat) made visual contact with the aircraft while my student continued to fly the approach. I communicated to the Pilot Flying that the other traffic did indeed appear to be intersecting.Shortly after; we received a traffic advisory from ZZZ Approach about the overtaking traffic; followed by instruction to contact ZZZ Tower. By this point the ADS-B contact was showing 100 feet below us and still overtaking/intersecting. I looked back but could no longer see the traffic. I read back our instruction and advised Approach that I had had that traffic in sight previously; but had lost him. I then called control of the aircraft and executed a right 360 turn to evade the overtaking traffic during the process of that turn I was able to reacquire visual on the overtaking traffic and it appeared that if I had not turned away he very likely might have collided with us.We made contact with ZZZ Tower; reported the 360 and why we were making it; and we received instruction to rejoin the approach at ZZZZZ. We did see and completed the approach without further incident. We did hear the pilot of the overtaking aircraft contact ZZZ tower at ZZZZZ. ZZZ Tower informed them of the near-miss and we heard the individual say he 'never saw us'.