Understanding nonequilibrium electron dynamics in a tunable-barrier dynamic quantum dot is vital to achieve high-accuracy single-electron pumping, which has applications to metrological standards and quantum information devices. However, the dynamic mechanism determining the pumping process has not been fully understood. In this paper, we study and clarify the physical mechanisms of single-electron pumping by analyzing and calculating master equations with a realistic model in detail. Our focus is mainly on the regime which lies at the crossover between the two extreme physical models of single-electron pumping. The mechanism crossover strongly depends on a capacitive coupling between a gate for tuning the barrier and the quantum dot. We find that there is an optimal value of the coupling with the best pumping accuracy and the mechanism crossover depends on gate voltage in some range of the coupling values. Our results offer a guideline for evaluation of pumping characteristics and for optimization of the device structure, which is an important step toward understanding high-speed nonequilibrium electron dynamics and realizing reproducible high-accuracy single-electron pumps.