Front dynamics with delays in a spatially extended bistable system of the reaction-diffusion type is studied by the use of nonlinear partial differential equation (PDE) of the parabolic type. The response of the self-ordered front, joining two steady states of the different stability in the system, to the multi-harmonic (step-like) force is examined. The relaxation rate of the system, that characterizes the delayed response of the front to the alternating current (ac) drive, is found to be sensitive to the peculiarities (shape) of the rate function (nonlinearity) of the governing PDE. By using computer simulations of the drift motion of the ac driven bistable front (BF) we are able to show that the characteristic relaxation time of the system decreases with the increasing outer slope parameters of the rate function and is not sensitive to the inner one.