  Geometric grid classes and the substitution decomposition have both been
shown to be fundamental in the understanding of the structure of permutation
classes. In particular, these are the two main tools in the recent
classification of permutation classes of growth rate less than
$\kappa\approx2.20557$ (a specific algebraic integer at which infinite
antichains begin to appear). Using language- and order-theoretic methods, we
prove that the substitution closures of geometric grid classes are partially
well-ordered, finitely based, and that all their subclasses have algebraic
generating functions. We go on to show that the inflation of a geometric grid
class by a strongly rational class is partially well-ordered, and that all its
subclasses have rational generating functions. This latter fact allows us to
conclude that every permutation class with growth rate less than $\kappa$ has a
rational generating function. This bound is tight as there are permutation
classes with growth rate $\kappa$ which have nonrational generating functions.
