During the first part of the training flight in the conventional wheel (tail wheel-equipped) airplane, the pilot receiving instruction made four stop-and-go landings on a runway that was almost directly aligned with the oncoming wind. At that point the instructor pilot took over in order to demonstrate a landing on a different runway where there was a crosswind of about 10 to 12 knots at 90 degrees to the runway. The instructor pilot made a successful touchdown on the main landing gear wheels, but as the airplane slowed and he began to lower the tail to the runway surface, the upwind wing was lifted by a gust of wind. Although the instructor pilot made control inputs to counter the lifting of the wing, the opposite wing impacted the surface of the runway, and the airplane veered toward the upwind runway edge. Before the instructor pilot could regain directional control, the airplane departed the right side of the runway, entered an area of soft terrain, and nosed over onto its back. According to the instructor pilot, there was no evidence of a flight control or rudder control malfunction.