  Ehrenfeucht-Fra\"iss\'e games provide means to characterize elementary
equivalence for first-order logic, and by standard translation also for modal
logics. We propose a novel generalization of Ehrenfeucht- Fra\"iss\'e games to
hybrid-dynamic logics which is direct and fully modular: parameterized by the
features of the hybrid language we wish to include, for instance, the modal and
hybrid language operators as well as first-order existential quantification. We
use these games to establish a new modular Fra\"iss\'e-Hintikka Theorem for
hybrid-dynamic propositional logic and its various fragments. We study the
relationship between countable game equivalence (determined by countable
Ehrenfeucht- Fra\"iss\'e games) and bisimulation (determined by countable
back-and-forth systems). In general, the former turns out to be weaker than the
latter, but under certain conditions on the language, the two coincide. We also
use games to prove that for reachable image-finite Kripke structures elementary
equivalence implies isomorphism.
