A finite morphism between stable marked curves over a non-archimedean field gives rise in a natural way to a morphism of metric graphs (also known as "abstract tropical curves").  We study the question of which morphisms between abstract tropical curves arise from this construction.  This leads us naturally to the notion of metrized complexes of curves (which are tropical curves endowed with some extra structure) and of harmonic morphisms between metrized complexes.  We show using Berkovich's theory of analytic spaces that every tamely ramified finite harmonic morphism of metrized complexes of curves arises from a morphism of algebraic curves.  This generalizes and provides new analytic proofs of earlier results of Saidi and Wewers.  We also present counterexamples to some possible strengthenings of this result; for example, the gonality of a tropical curve can be strictly smaller than the gonality of any smooth proper lift.  As an application of the above considerations, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Neron models, providing a negative answer to a question of Ribet.