We study locally interacting processes in discrete time, often called probabilistic cellular automata, indexed by locally finite graphs. For infinite regular trees and certain generalized Galton-Watson trees, we show that the marginal evolution at a single vertex and its neighborhood can be characterized by an autonomous stochastic recursion referred to as the local-field equation. This evolution can be viewed as a nonlinear or measure-dependent chain, but the measure dependence arises from the symmetries of the underlying tree rather than from any mean field interactions. We discuss applications to simulation of marginal dynamics and approximations of empirical measures of interacting chains on several generic classes of large-scale finite graphs that are locally tree-like. In addition to the symmetries of the tree, a key role is played by a second-order Markov random field property, which we establish for general graphs along with some other novel Gibbs measure properties.