We study higher-spin ($S\ensuremath{\ge}1$) generalization of the one-dimensional Kondo-Heisenberg model, in which the local spin-$S$ moments of the Kondo-lattice model interact with each other via the antiferromagnetic Heisenberg interaction (${J}_{\text{H}}$), by analytical and numerical methods. The strong-coupling (i.e., large Kondo coupling) expansion maps out an insulating phase at half-filling whose magnetic correlation depends on the parity of $2S$ as well as a ferromagnetic metallic phase which dominates the strong-coupling region at generic fillings. Then, we carried out the density-matrix renormalization-group simulations for $S=1$ to closely investigate the phase structure at large but finite Kondo coupling. At half-filling, the Kondo coupling and ${J}_{\text{H}}$ do not compete and the insulating spin-gapless phase is stable, while the competition of the two leads to a stepwise collapse of the strong-coupling ferromagnetism via an intervening dimerized insulating phase with power-law spin correlation at quarter-filling.