Flight Instructor with student reported engine smoke; loss of power; and a subsequent engine failure in a DA40. The Instructor performed a power-off; safe landing at a nearby airport.

Narrative

Original plan to fly to ZZZ [airport] for Landings and Maneuvers in the airspace for a refresher flight for a student. After flight lesson elements were completed we climbed to 3500 for the arrival. Upon reaching altitude; performing checklist; and checking in with ATC we realized our RPM's and Load was oscillating from 75% to 85% load. We also began to smell smoke. There was no annunciations or Alerts to advise us something was wrong. Once we identified the abnormalities a positive exchange of flight controls was initiated by the instructing pilot and autopilot was disconnected and the instructing pilot flew the aircraft and handled coms while the student pilot ran the QRC/QRH. We [requested priority handling] with ZZZ ATC and chose to divert to ZZZ since we were approximately 5 nm to the north of the airport. ZZZ ATC approved of this and told us to squawk VFR; frequency change approved and 2 planes in the pattern at ZZZ". Prior to the descent we initiated the engine trouble shooting in flight checklist and upon giving the aircraft max load; we lost about 50% of our power. After identifying the loss of power the instructing pilot realized that the load was not able to sustain power lever changes and when increasing the power we would lose the ability to go back to previous power settings. During the descent down to ZZZ the instructing pilot [advised] on the ZZZ unicom while giving traffic advisories to planes in the pattern. Traffic in the pattern at ZZZ was on the upwind and downwind of Runway XX. They extended both of their legs and allowed us to get in as a priority aircraft. During the descent down we lost positive engine control and the instructing pilot performed a power off (power level was at idle) landing from base to final at approximately 1300 feet msl. Landing was made safely and propeller continued to oscillate fluctuating from 1% load to about 11% with RPMs fluctuating from approximately 500-900. Oil pressure 1.4 and unstable. We were able to taxi and park on a parking tee where we advised traffic at ZZZ we were clear of the runway and attempted to contact ATC; secured the aircraft; and exited the plane as it was smoking badly and in the moment we did not know where the smoke was coming from. Once on the ground we contacted Flight ops and Maintenance. Once Maintenance arrived and did their initial inspection. The belly of the aircraft was covered in oil from gascolator to tail and inside of the cowling the aircraft was virtually clean. Upon further inspection Multiple possible failures were identified with the turbocharger and oil lines."