  We consider four-dimensional $\mathcal{N}=1$ supergravity models of a kind
appearing in string flux compactifications. It has been recently shown that, by
using double three-form multiplets instead of ordinary chiral multiplets, one
can promote to dynamical variables (part of) the quantized numbers appearing in
the flux-induced superpotential. We show that double three-form multiplets
naturally transform under symplectic dualities associated with the special
K\"ahler structure that characterizes their scalar sector. Furthermore, we
discuss how to couple membranes which carry arbitrary `electric-magnetic'
charges. The complete action is supersymmetric, kappa-symmetric and duality
covariant. As an application, we derive the flow equations for BPS domain walls
sourced by membranes and give simple analytic examples of their solution.
