This study investigates the impact of sliders – constraints acting on elastic rods allowing for a transverse displacement jump while maintaining axial and rotational displacement continuity – on the dynamics of a periodic elastic grid, including the effects of axial preload. The grid is linearly elastic and subject to in-plane incremental deformation, involving normal and shear forces and bending moment. The periodicity of the infinite grid permits a Floquet–Bloch wave analysis and a rigorous dynamic homogenization, leading to an equivalent prestressed elastic solid. The investigation is complemented by ad hoc developed F.E. simulations and perturbations with a pulsating Green’s function. Results show that the sliders create band gaps, flat bands and Dirac cones in the dispersion diagrams and generate macro-instability even for tensile prestress. The latter corresponds to the loss of ellipticity at the parabolic boundary in the equivalent elastic solid and provides a rare example of an almost unexplored form of material instability. Therefore, our results offer design strategies for metamaterials and architected materials showing reversible material instabilities and filtering properties for mechanical signals.