I have been unable to find a general definition of semantic consequence ($\vDash$) for modal logic, so I would appreciate if you commented on this speculation of mine:
Definition. Let $M = (W, R, V)$ be a model of modal logic and $X$ a set of formulas. Then, $$ X \vDash \varphi :\Leftrightarrow \forall M.\forall w \in W.(\forall\psi \in X.(M\vDash_w \psi) \Rightarrow M \vDash_w \varphi). $$
Here, $M \vDash_w \varphi$ is the standard recursively defined satisfiability relation.
Did I get it right? Thank you for the feedback. :)
P.S. For added context, I am trying to formalize the below definition found in the SEP entry on modal logic. That is, my definition of $\vDash$ is really only a formalization of $\mathbf{K}$-validity.
There exist two definitions of consequence in modal logic:
local validity: Truth is preserved on a per-world level; this is the definition you are proposing.
global validity: Truth is preserved at a per-model level: $X \vDash \phi \Leftrightarrow \forall M. ((\forall \psi \in X. M \models \psi) \to M \models \phi)$, i.e., $\Leftrightarrow \forall M. ((\forall \psi \in X. \forall w \in W. M \models_w \psi) \to (\forall w \in W. M \models_w \phi$)).
The local definition is the one that is probably the more convincing one intuitively.