The overvoltage and surge energy robustness of GaN power high-electron-mobility transistors (HEMTs) is a key gap recently identified by the JEDEC JC-70.1 committee. This work presents the first study on the parametric shift and recovery of an industrial depletion-mode metal-insulator-semiconductor HEMT (MIS-HEMT) in repetitive overvoltage switching close to its dynamic breakdown voltage. In each switching cycle, a voltage overshoot of up to 90% of dynamic breakdown voltage was applied during the device turn-OFF process. As the repetitive switching prolongs, the device showed shifts in threshold voltage, saturation current, and on-resistance, and these parametric shifts saturated after 1-million cycles. These parametric shifts are believed to be induced by the trapping of the holes generated in the impact ionization. After the repetitive switching, the device exhibited slow recovery in the natural state, at elevated temperatures, or with negative gate biases, due to the difficulty in hole removal. By contrast, applying a positive gate bias or a positive substrate bias can facilitate the hole migration towards the two-dimensional-gas (2DEG) channel for recombination, and therefore, accelerated the device recovery. In particular, a 50-V substrate bias allowed the device to recover in a few minutes. This work shows the good overvoltage robustness of GaN MIS-HEMTs and unveils effective methods for their post-switching recovery, as well as suggests the significance of hole dynamics for the device overvoltage switching close to the dynamic breakdown voltage.