In this paper we introduce natural metrics in the hyperbolic α-Bloch and hyperbolic general Besov-type classes F∗(p, q, s). These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, compact composition operators Cφ acting from the hyperbolic α-Bloch class to the class F∗(p, q, s) are characterized by conditions depending on an analytic self-map φ : D → D.