  We consider uniform random permutations in classes having a finite
combinatorial specification for the substitution decomposition. These classes
include (but are not limited to) all permutation classes with a finite number
of simple permutations. Our goal is to study their limiting behavior in the
sense of permutons.
  The limit depends on the structure of the specification restricted to
families with the largest growth rate. When it is strongly connected, two cases
occur. If the associated system of equations is linear, the limiting permuton
is a deterministic $X$-shape. Otherwise, the limiting permuton is the Brownian
separable permuton, a random object that already appeared as the limit of most
substitution-closed permutation classes, among which the separable
permutations. Moreover these results can be combined to study some non strongly
connected cases.
  To prove our result, we use a characterization of the convergence of random
permutons by the convergence of random subpermutations. Key steps are the
combinatorial study, via substitution trees, of families of permutations with
marked elements inducing a given pattern, and the singularity analysis of the
corresponding generating functions.
