There is a 'research report(?)' from the Institute for Logic, Language, and Computation (a rersearch institute of the University of Amsterdam) by Goivanna D'Agostino, Angelo Montanari, and Alberto Policriti titled "Modal Logic and Set Theory: a Set-Theoretic Interpretation of Modal Logic" (look under title on the Web). This report discusses the very topic (the set-theoretic interpretation of modal system $K$) of your question. I hope it is helpful to you.

There is a 'research report(?)' from the Institute for Logic, Language, and Computation (a rersearch institute of the University of Amsterdam) by Goivanna D'Agostino, Angelo Montanari, and Alberto Policriti titled "Modal Logic and Set Theory: a Set-Theoretic Interpretation of Modal Logic" (look under title on the Web). This report discusses the very topic (the set-theoretic interpretation of modal system $K$) of your question. I hope it is helpful to you.

You might also want to take a look at "Modal Deduction in Second-Order Logic and Set Theory--$I$" by the aforementioned authors and Johan van Benthem. It also discusses the set-theoretic interpretation of system $K$.