The kink dynamics of the damped Frenkel-Kontorova (discrete sine-Gordon) chain driven by a constant external force is investigated. Resonant steplike transitions of the average velocity occur due to the competition between the moving kinks and their radiated phasonlike modes. A mean-field consideration is introduced to give a precise prediction of the resonant steps. Slip-stick motion and spatiotemporal dynamics on those resonant steps are discussed. Our results can be applied to studies of the fluxon dynamics of one-dimensional Josephson-junction arrays and ladders, dislocations, tribology, and other fields.