How does the problem of quantified modal logic relate to Kripke's model

Question

Throughout multiple papers and reviews, Kripke's models are viewed as a countertheory to the work of Quine on the three grades of modal involvement. I understand how the modal is set up: I consider a Kripke Modal as a First-Order Model with the addition of worlds and its relationship. I can also conceive how this in general builds a structure for quantified modal logic. However, I cannot see how such a model views a problem like a problem to which Quine usually refers to: \begin{align*} "(\exists x)(\text{nec}(x > 9))" \end{align*} or the problem with distinguishing \begin{align} "\text{nec}(9 > 5)" \\ "\text{nec(#planets} > 5)" \end{align} where the first $9>5$ statement seems true and the second statement seems to be false. Does a Kripke-Model provide a framework for this problem?