  We define a finite-field version of Appell-Lauricella hypergeometric
functions built from period functions in several variables, paralleling the
development by Fuselier, et. al in the single variable case. We develop
geometric connections between these functions and the family of generalized
Picard curves. In our main result, we use finite-field Appell-Lauricella
functions to establish a finite-field analogue of Koike and Shiga's cubic
transformation for the Appell hypergeometric function $F_1$, proving a
conjecture of Ling Long. We use our multivariable period functions to construct
formulas for the number of $\mathbb{F}_p$-points on the generalized Picard
curves. We also give some transformation and reduction formulas for the period
functions, and consequently for the finite-field Appell-Lauricella functions.
