Edney’s Type I and Type II shock interactions between an incident shock and a bow shock generated by a falling/rising cylinder are investigated at a freestream Mach number of 6. The numerical results indicate that the transition from Type I to Type II occurs with the cylinder falling, whereas the transition from Type II to Type I occurs with the cylinder rising. The transition boundaries are sensitive to the cylinder velocities and hysteresis phenomena are observed. Compared with the transition boundaries of stationary cylinder, the intersection point between the incident shock and bow shock for falling cylinder is upward, whereas that for rising cylinder is downward. The hysteresis loop enlarges with the increase in the absolute value of the cylinder velocity. The changes in the curvature of the bow shock and the deformation of the subsonic region behind the bow shock in the dynamic cases of falling/rising cylinder are responsible for the transition hysteresis.