In this paper, the role of gradient micro-inertia termsη¯∥ ∇u,t∥2and free micro-inertia termsη∥P,t∥2is investigated to unveil their respective effects on the dynamic behaviour of band-gap metamaterials. We show that the termη¯∥ ∇u,t∥2alone is only able to disclose relatively simplified dispersive behaviour. On the other hand, the termη∥P,t∥2alone describes the full complex behaviour of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows us to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertiaη¯∥ ∇u,t∥2, in the sense of Cartan–Lie decomposition of matrices, allows us to flatten separately the longitudinal and transverse optic branches, thus giving us the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertiaη¯∥ ∇u,t∥2on more classical enriched models such as the Mindlin–Eringen and the internal variable ones. We find that the addition of such a gradient micro-inertia allows for the onset of one band-gap in the Mindlin–Eringen model and three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for, which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the best performing one, among the considered enriched models, for the description of non-local band-gap metamaterials.