Given a sequence of regular graphs G_n whose size goes to infinity, and dynamics that are suitably symmetric, a key question is to understand the limiting dynamics of a typical particle in the system. The case when each G_n is a clique falls under the purview of classical mean-field limits, and it is well known that (under suitable assumptions) the dynamics of a typical particle is governed by a nonlinear Markov process.  In this talk, we consider the complementary sparse case when G_n converges in a suitable sense to a countably infinite locally finite graph G, and describe various limit results, both in the setting of diffusions and Markov chains.  In particular, when G is a d-regular tree, we obtain an autonomous characterization of the local dynamics of the neighborhood of a typical node. We also obtain a local characterization for the annealed dynamics on a class of Galton-Watson trees.   The proofs rely on a certain Markov random field structure of the dynamics on the countably infinite graph G, which may be of independent interest.   This is based on various joint works with Ankan Ganguly, Dan Lacker, Mitchell Wortsman and Ruoyu Wu.