  In this article we continue the study of holonomic modules over sheaves of
Cherednik algebras, initiated by the third author in [Tho18]. Working with
arbitrary parameters, we first develop a theory of $b$-functions to prove that
push-forward along open embeddings preserves holonomicity. This implies that
pull-back along closed embeddings also preserves holonomicity. We use these
facts to show that both push-forward and pull-back under any melys morphism
preserves holonomicity. Since duality preserves holonomicity, we deduce that
extraordinary push-forward and extraordinary pull-back also exist for holonomic
modules.
  As a consequence, we give a general classification of irreducible holonomic
modules similar to the classification of irreducible holonomic
$\mathscr{D}$-modules as minimal extensions of integrable connections on
locally closed subsets. Finally, we prove that ext-groups between holonomic
modules are finite-dimensional and explore applications of our work to the
classification of aspherical parameters and existence of finite-dimensional
modules for sheaves of Cherednik algebras.
