Antimony sulfide (Sb2S3) and selenide (Sb2Se3) are emerging earth-abundant absorbers for photovoltaic applications. Solar cell performance depends strongly on charge-carrier transport properties, but these remain poorly understood in Sb2X3 (X = S, Se). Here we report band-like transport in Sb2X3, determined by investigating the electron-lattice interaction and theoretical limits of carrier mobility using first-principles density functional theory and Boltzmann transport calculations. We demonstrate that transport in Sb2X3 is governed by large polarons with moderate Fröhlich coupling constants (α ≈ 2), large polaron radii (extending over several unit cells), and high carrier mobility (an isotropic average of >10 cm2 V-1 s-1 for both electrons and holes). The room-temperature mobility is intrinsically limited by scattering from polar phonon modes and is further reduced in highly defective samples. Our study confirms that the performance of Sb2X3 solar cells is not limited by intrinsic self-trapping.