Transient ignition process of premixed stagnation-point flows over a catalytic surface of a solid plate with a finite thickness is investigated numerically in this work. The results reveal that the thermal runaway criterion instead of the zero-gradient criterion is preferred for the problem of interest. Depending on system parameters, both the ignition delay and the critical rate of catalytic reactions at ignition are either conductively or catalytically controlled. The effects of catalytic reactions on ignition are positive and negative for catalytically and conductively controlled ignition mechanisms, respectively. According to these two ignition mechanisms, the qualitative and quantitative results of the ignition delay and the critical rate of catalytic reactions at ignition are systematically analyzed. In particular, the minimum ignition delay and the C-shaped ignition curve are discussed.