  We consider uniform random permutations in proper substitution-closed classes
and study their limiting behavior in the sense of permutons.
  The limit depends on the generating series of the simple permutations in the
class. Under a mild sufficient condition, the limit is an elementary
one-parameter deformation of the limit of uniform separable permutations,
previously identified as the Brownian separable permuton. This limiting object
is therefore in some sense universal. We identify two other regimes with
different limiting objects. The first one is degenerate; the second one is
nontrivial and related to stable trees.
  These results are obtained thanks to a characterization of the convergence of
random permutons through the convergence of their expected pattern densities.
The limit of expected pattern densities is then computed by using the
substitution tree encoding of permutations and performing singularity analysis
on the tree series.
