There are a number of different models for defining transformations between languages. Finite state transducers and MSO-definable graph transformations over string graphs are the two that I am best acquainted with. We know that 2-way finite state transducers (which are more expressive than their 1-way counterparts) and MSO-definable string transformations capture the same set of transformations along with some other less well-known models that use combinators. This class of transformations is considered regular, and so it is easy to show then that a transformation is regular if you can provide a description of it with one of these models.
Is there a straight-forward way to say that a transformation is outside of this class? Something akin to the pumping lemma for regular languages or the Myhill-Nerode theorem but for string transformations is the sort of thing I am looking for.