The post-breakdown (BD) current-voltage (I-V) characteristics in MgO/Si and MgO/InP stacks with metal gate were investigated. We show that both stacks exhibit the soft and hard BD conduction modes and that the magnitude of the post-BD currents depends statistically on the substrate material, being larger in the case of the InP samples. This is contrary to what happens with the current in the fresh devices, which irrespective of the InP surface treatment, is larger for the Si samples. A comparative analysis of the post-BD conduction characteristics and direct evidences of the localized thermal effects on the metal gate electrode of both structures are presented.