  We study the relationship between tropical and classical Hurwitz moduli
spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a
tropicalization for the moduli space of generalized Hurwitz covers of an
arbitrary genus curve. Our approach is to appeal to the geometry of admissible
covers, which compactify the Hurwitz scheme. We define and construct a moduli
space of tropical admissible covers, and study its relationship with the
skeleton of the Berkovich analytification of the classical space of admissible
covers. We use techniques from non-archimedean geometry to show that the
tropical and classical tautological maps are compatible via tropicalization,
and that the degree of the classical branch map can be recovered from the
tropical side. As a consequence, we obtain a proof, at the level of moduli
spaces, of the equality of classical and tropical Hurwitz numbers.
