Finite state transducer with infinitary outputs or without emphasis on acceptance?
There are several relevant references on the subject. I just indicate a few of them. I suggest you to browse the references of these articles to get more.

Early references include the work of Françoise Gire.

[1] F. Gire, "Relations rationnelles infinitaires", Thèse de 3ème cycle, Paris VII, 1981.

[2] F. Gire, Une Extension aux Mots Infinis de la Notion de Transduction Rationnelle, 6th GI Conf., Lect. Notes in Comp. Sci., Volume 145, 1983, p. 123-139.

[3] F. Gire, M. Nivat, Relations rationnelles infinitaires. [Infinitary rational relations] Calcolo 21 (1984), no. 2, 91--125.

Interesting results can be found in

[4] Latteux, M.; Timmerman, E. Rational $\omega$-transductions. Mathematical foundations of computer science (Banská Bystrica, 1990), 407--415, Lecture Notes in Comput. Sci., 452, Springer, Berlin, 1990.

Section 4.3 of [5] is also a very good reference.

[5] L. Staiger, ω-Languages, Vol 3, Chapter 6 of the Handbook of Formal languages edited by G. Rozenberg and A. Salomaa, (1997) 339-387. Springer-Verlag, Berlin.

Finally, here is a more recent paper, where you will find other relevant references.

[6] Finkel, Olivier. On the accepting power of 2-tape Büchi automata. STACS 2006, 301--312, Lecture Notes in Comput. Sci., 3884, Springer, Berlin, (2006).