In nonlinear periodic systems, the relation between nonlinearity and dispersion can generate spatially localized band-gap solitons. One of the outstanding features of these solitons is that they are practically immobile. In this study, we propose a means of spatially moving band-gap solitons through the periodic modulation of periodic potentials. We first numerically identify stable band-gap solitons, then set them as initial states and allow them to evolve with the modulation of the relevant phases of periodic potentials. We find that if the modulation is sufficiently slow, the band-gap solitons can be transported away at a very high fidelity. This work provides a possible theoretical scheme for transporting band-gap solitons.