In the comments below this answer, there is a question about whether "the theory" of a certain proper class is strictly weaker than Kripke-Platek. Now, if this were a set rather than a proper class, I would have immediately said, "it is a complete theory, so it can't be strictly weaker than anything (in a direct sense, anyway)". But it is a proper class, so there is no real notion of its complete theory, and I realized I don't know if there is any standard notion of the theory of a proper class. For instance, there is a reasonably natural sense in which $\mathsf{ZF+V=L}$ is "the theory of $L$": $$\mathsf{ZF}\vdash \varphi^L\iff \mathsf{ZF+V=L}\vdash \varphi.$$ On the other hand, $\mathsf{ZF+V=HOD}$ is not the theory of $HOD$ in this sense, although perhaps there is another theory that is.
So my question is whether there is some well-established idea of a theory of a proper class. Even for set-sized models, we could have a notion of the sentences that ZF(C) can prove are true or false in the model. The question is whether this is well-established and useful for anything in particular.