  Let $f$ be a postcritically finite branched self-cover of a 2-dimensional
topological sphere. Such a map induces an analytic self-map $\sigma_f$ of a
finite-dimensional Teichm\"uller space. We prove that this map extends
continuously to the augmented Teichm\"uller space and give an explicit
construction for this extension. This allows us to characterize the dynamics of
Thurston's pullback map near invariant strata of the boundary of the augmented
Teichm\"uller space. The resulting classification of invariant boundary strata
is used to prove a conjecture by Pilgrim and to infer further properties of
Thurston's pullback map. Our approach also yields new proofs of Thurston's
theorem and Pilgrim's Canonical Obstruction theorem.
