We study a class of Hamiltonian partial differential equations introduced by Bridges (2006) in space and space-time. These systems satisfy a multisymplectic conservation law, which can be viewed as a multidimensional generalization of the symplectic properties inherent in Hamiltonian ODEs. A central question in structure-preserving discretization is determining which finite element methods maintain this property at the discrete level. To address this, we introduce a unified hybridization framework that allows for the simultaneous analysis of local conservation properties across diverse discretization families. We demonstrate that while many methods, such as certain hybridizable discontinuous Galerkin (HDG) schemes, satisfy a strong version of multisymplecticity, the standard conforming finite element exterior calculus (FEEC) methods of Arnold, Falk, and Winther (2006) satisfy this property only in a weak sense. Finally, we establish the bridge between this local multisymplectic perspective and the classical treatment of PDEs as infinite-dimensional Hamiltonian ODEs. This is a joint work with Ari Stern (WashU).