We describe a mathematical framework for equational reasoning about infinite families of string diagrams which is amenable to computer automation. The framework is based on context-free families of string diagrams which we represent using context-free graph grammars. We model equations between infinite families of diagrams using rewrite rules between context-free grammars. Our framework represents equational reasoning about concrete string diagrams and context-free families of string diagrams using double-pushout rewriting on graphs and context-free graph grammars respectively. We prove that our representation is sound by showing that it respects the concrete semantics of string diagrammatic reasoning and we show that our framework is appropriate for software implementation by proving the membership problem is decidable.