Rational Functions and CFL
In general the determinism of the transducer seems to be no real restrition. The choices that can be made by a transducer can be moved into the input language. If that is OK for you, the objects sometimes denoted ${\cal M}(L)$ are called principal trio's. Examples include families of linear, blind counter and one turn counter languages (fixed number of counters).
The Habilitation-thesis of Kluas Reinhardt (Counting as Method, Model and Task in Theoretical Computer Science, available on-line) contains a nice inclusion diagram of several families (page 64).