In the current work, we develop a higher gradient dynamic homogenization method with micro-inertia effects. To that scope, we compute the macroscopic constitutive parameters up to the second gradient, using two distinct approaches, namely the Hamilton's principle and the total internal energy formulation. Thereupon, we analyze the sensitivity of the second gradient constitutive terms on the inner material and geometric parameters for the case of composite materials with a periodic, layered microstructure. The results suggest that the significance of the second gradient terms highly depends on the differences in the geometric and material properties of the underlying microstructural phases, as well as on the deformation mode of interest. What is more, the wave propagation study indicates that different higher gradient results are obtained, depending on the formulation used and on the wavenumber range of interest. In particular, Hamilton based, second gradient models perform poorly in the low wavenumber range, compared to first gradient or to analytic, exact solutions. Contrariwise, second gradient macroscopic constitutive formulations derived using the total higher gradient internal energy outperform first gradient approaches, better approximating the wave propagation characteristics of the effective medium, as the corresponding comparison in between the frequency diagrams and phase and group velocities suggest.