We have investigated the switching dynamics of the first and second switches in intrinsic Josephson junctions (IJJs) of Bi₂Sr₂CaCu₂O with different maximum Josephson current density J _c to reveal the doping evolution of interaction between IJJs. For the second switch, the crossover temperature between temperature-independent switching similar to quantum tunneling and thermally activated switching  is remarkably higher than that for the first switch. Moreover,  slightly decreases with increasing J _c, which violates the conventional relation between the crossover temperature and the critical current density. These features can be explained not by a change in the Josephson coupling energy but by a change in the charging energy of the Josephson junction. We argue that the capacitive coupling model explains the increase in the fluctuation in the quantum regime of the second switch and the anti-correlation between  and J _c. Furthermore, inductive coupling does not contribute to these peculiar phenomena in the switching dynamics of stacked IJJs.