We investigate charge transport in one-dimensional arrays of Josephson junctions. In the interesting regime of ``small charge solitons'' (polarons), $\ensuremath{\Lambda}{E}_{J}>{E}_{C}>{E}_{J}$, where $\ensuremath{\Lambda}$ is the (electrostatic) screening length, the charge dynamics are strongly influenced by the polaronic effects (i.e., by dressing of a Cooper pair by charge dipoles). In particular, the soliton's mass in this regime scales approximately as ${E}_{J}^{\ensuremath{-}2}$. We employ two theoretical techniques: the many-body tight-binding approach and the mean-field approach, and the results of the two approaches agree in the regime of ``small charge solitons.'' Renormalization of the soliton's mass could be observed; for example, as enhancement of the persistent current in a ring-shaped array.