The work focuses on the dynamic continualization of periodic materials to obtain thermodynamically consistent models that accurately describe the dispersion features of the corresponding Lagrangian/heterogeneous systems. Equivalent non-local integral and gradient-type higher-order continuum models are obtained through an enhanced continualization of the governing equations of the starting model [1]. Homogeneous nonlocal continuum models of increasing order are formulated using a formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions accounting for shift operators and proper pseudo-differential downscaling laws [2, 3]. They are characterized by energetically consistent differential equations with inertial and constitutive non-localities. Different systems have been considered: from multilayered antitetrachiral elastic lattice materials with lumped masses at the nodes and local resonators to visco-electro-elastic periodic laminates. In any case, an excellent agreement has been obtained between the dispersion curves of the discrete/heterogeneous systems and the ones of the obtained gradient-continuum models [4,5].