To bridge laboratory experiments and analytical theory, we studied by Monte Carlo simulations the kinetic properties of the reaction front formed under initially separated conditions in the A + B --> C reaction-diffusion system, for low reaction rates. We observed crossovers in (1) global reaction rates, (2) the position of the reaction front center, and (3) the width of the reaction front. The motion of the reaction front showed a U-turn in some cases but not in all the cases predicted by theory. Quantitative results, including time exponents, an presented and discussed. The very rich phenomenology of this simple elementary reaction is demonstrated and interpreted.