  We investigate the limiting behavior of solutions to the inhomogeneous
$p$-Laplacian equation $-\Delta_p u = \mu_p$ subject to Neumann boundary
conditions. For right hand sides which are arbitrary signed measures we show
that solutions converge to a Kantorovich potential associated with the geodesic
Wasserstein-$1$ distance. In the regular case with continuous right hand sides
we characterize the limit as viscosity solution to an infinity Laplacian /
eikonal type equation.
