PA-32 instructor with student reported a runway excursion after landing at a non-towered airport requiring the instructor to take over the controls to bring the aircraft back to the runway pavement. Instructor stated the student was new to the PA-32 which along with variable winds contributed to the momentary runway excursion.

Narrative

During a routine private pilot flight training mission; which involves transitioning to new aircraft during training; a runway excursion occurred at touchdown.The instructor and student pilot were practicing pattern entry and landings at ZZZ in order to take advantage of longer and wider runways; with less crowded conditions; early morning. The wind conditions were 45 degree left crosswind component of 5 to 7 knots; with gusts to 10 knots. Several landings were made during the instruction period prior to the incident being reported.On the 3rd or 4th landing; variable wind and wind gusts caused the student to over and under control the aircraft during the approach to landing and round out. The aircraft touched down to the left of the runway centerline; and while weathervaning into the crosswind. Upon touchdown; the aircraft veered in the direction of the wind; and the left main gear wheel left the pavement for a few seconds at landing speed. The aircraft wing traversed several runway edge lights without making any contact. The student and instructor reacted to bring the aircraft left wheel back onto the pavement and added normal braking to bring the aircraft to a normal rolling position before exiting to taxiway.The aircraft was taken to an adjacent ramp area for an examination of the left wing; aileron; flap; main gear and fuselage. No impact damage was found; and there weren't even any grass or grass stains discovered on the plane or tires.The instructor attributes the episode to improper pattern airspeed and altitude; higher than recommended final approach airspeed; wind conditions exceeding the student pilot capability; and inadequate supervision by the CFI onboard during the transition training.