Is there any field that studies the equivalent of cellular automata, but for arbitrary graphs rather than 1D cells? A cellular automaton consists of a sequence of cells, each with a state, and a globally symmetric transition rule based on the neighbors of a cell. They can be interpreted as graphs where each cell is a node, and each node has an edge to a neighbor.

Is there any field studying that system, but generalized to arbitrary graphs - i.e., allowing cells to have, for example, 3 or more neighbors?
Graph dynamical systems is a general term for systems where each vertex of a graph carries a state, and the system evolves in time so that the next state of a vertex only depends on the states in its neighborhood (possibly in a nondeterministic way). Boolean networks are a special case where the graph is finite and each state is either $0$ or $1$. Boolean networks are used e.g. as models of gene expression in cells, and a lot of practical and theoretical research exists on them.