We prove a recent conjecture arising in the context of scattering amplitudes for a `motivic' Galois group action on Gauss' ${}_2F_1$ hypergeometric function. More generally, we show on the one hand how the coefficients in a Laurent expansion of a Lauricella hypergeometric function can be promoted, via the theory of motivic fundamental groups, to motivic multiple polylogarithms. The latter admit a `local' action of the usual motivic Galois group. On the other hand, we define lifts of the full Lauricella functions as matrix coefficients in a Tannakian category of twisted cohomology, which inherit a `global' action of its Tannaka group. We prove that these two actions are compatible. We also study single-valued versions of these hypergeometric functions, which may be of independent interest.