Consider two logical languages $\mathcal L_1\subseteq\mathcal L_2$. Let the corresponding logics be semantically defined. Suppose that the correspond models depend on the language itself, but there is a meaningful way to convert the models associated with $\mathcal L_1$ into the others and vice versa. How can one compare expressive strength of such two logics if the model classes are not the same? Would one use the model class associated with the larger language?
The example I have in my mind comes from a paper about justification logic by Bryan Renee(pdf). In his paper, we compares a "basic" justification logic with one enhanced by a public announcement operator. The classical semantic way of defining justification logics is to consider extensions of Kripke-models where a function $$ \mathcal E:\mathcal L\times Jt\to \mathcal P(W) $$ is added to assign every formula and justification term a set of worlds where this combination is considered admissible. The classic definition about expressive comparison is considering two languages evaluated over the same class of models. But enhancing the language should also change the class of models as a part of the definition depends on it. Naturally, if one considers an extension of the language like Renee does, there is here also a way to convert/extend a model from the lower language to a model concerning the higher language. My confusion arises as how one would compare these logics then, would one use the "larger" model class? Renee makes no comment on this.