  We study the existence and uniqueness of mild and strong solutions of
nonlocal nonlinear diffusion problems of $p$-Laplacian type with nonlinear
boundary conditions posed in metric random walk spaces. These spaces include,
among others, weighted discrete graphs and $\mathbb{R}^N$ with a random walk
induced by a nonsingular kernel. We also study the case of nonlinear dynamical
boundary conditions. The generality of the nonlinearities considered allow us
to cover the nonlocal counterparts of a large scope of local diffusion
problems: Stefan problems, Hele-Shaw problems, diffusion in porous media
problems, obstacle problems, and more. Nonlinear semigroup theory is the basis
for this study.
