We try to present a systematic treatment of equivalence problems concerning deterministic generalized sequential machines (gsm), functional transducers and deterministic transducers as well as compositions of functional transducer mappings, especially morphisms, and their inverses. It is assumed that the language families in question are effectively closed under inverse morphism and intersection with regular languages. If ℒ is such a family and if the morphism equivalence problem for it is decidable, then also the deterministic gsm equivalence problem for it is decidable. If, in addition, the emptiness problem is decidable for ℒ, then the problem of transducers functional equivalence and the deterministic transducer equivalence problem are decidable for ℒ. In applications concerning finite test sets special attention is paid to the supports of rational formal power series. On the equivalence and length equivalence of compositions of a morphism and an inverse morphism several undecidability results are pre...