Abstract: Dynamical systems whose number and connectivity of state variables evolve over time can often be modeled by spatially embedded graph-local dynamics expressible by rewrite rules in a “dynamical graph grammar” (DGG). We mention examples in biology (multicellular tissues and cytoskeletal polymer networks) and materials science (dislocations, force chains, fractures), but a wide variety of emergent phenomena may be so described, alternatively to particle and field models. We describe a Master Equation operator algebra framework for defining DGG dynamics including stochastic and ordinary differential equation dynamics; sketch how several algorithms for simulation and machine-learned model reduction can result; and point out a number of open areas for mathematical research that would help to push forward the scientific applications. These open areas include exploiting the operator algebra for solvable subsystems; DGG-centered model reduction and analysis methods; the incorporation of PDE dynamics for fields and strata; the dynamics of more abstract rewrite rules; specializing simulation and learning algorithms for substantial gains in efficiency; and using interactive theorem verification to systematize such a flexible approach to scientific modeling.