if $\kappa$ is a cardinal such that $V_\kappa \models \sf ZFC$, then $\kappa$ is called a worldly cardinal and this is not necessarily downward absolute [Hamkins], i.e. there is a model $V$ of $\sf ZFC$ which satisfies the existence of a worldly cardinal $\kappa$, and there is a subset $W$ of $V$ that is a transitive inner model of $\sf ZFC$ and yet doesn't satisfy $\kappa$ being a worldly cardinal.

Would that phenomena repeat itself upwardly beyond inaccessibility? I mean for example the worldly cardinak $\kappa$ such that $V_\kappa \models (\sf ZFC+ \text {there is an inaccessible})$ would that similarily be not downward absolute for that theory?