On the basis of local nonequilibrium approach the dynamics of a motion of the rapid solidification front in a binary alloy is considered. Asymptotic expansion for concentration of solute and its diffusion Aux at the front is found in the limit of large times t -> infinity. A selection criterion for dynamic stability of the front motion in a steady-state regime (V = V-o = const) is formulated. An expression for the definition of a nonstationary time during the transition to the steady-state regime is obtained. The front motion regimes with decreasing velocity in time are investigated. In these regimes the velocity decreases in time as V(t) similar to t(-1/2) and V(t) similar to t(-1/3).