We study the dynamical state of cores by using a simple analytical model, a sample of observational massive cores, and numerical simulations of collapsing massive cores. From the analytical model, we find that, if cores are formed from turbulent compressions, they evolve from small to large column densities, increasing their velocity dispersion as they collapse. This results in a time evolution path in the Larson velocity dispersion-size diagram from large sizes and small velocity dispersions to small sizes and large velocity dispersions, while they tend to equipartition between gravity and kinetic energy.From the observational sample, we find that: (a) cores with substantially different column densities in the sample do not follow a Larson-like linewidth-size relation. Instead, cores with higher column densities tend to be located in the upper-left corner of the Larson velocity dispersion σ v,3D -size R diagram, a result predicted previously (Ballesteros-Paredes et al. 2011a). (b) The data exhibit cores with overvirial values.Finally, in the simulations of collapsing cores we reproduce the behavior predicted by the analytical model and depicted in the observational sample: cores evolve towards larger velocity dispersions and smaller sizes as they collapse and increase their column density. More importantly, however, is that collapsing cores appear to approach overvirial states within a free-fall time. We find that the cause of this apparent excess of kinetic energy is an underestimation of the actual gravitational energy, due to the assumption that the gravitational energy is given by the energy of an isolated sphere of constant column density. We find that this apparent excess disappears when the gravitational energy is correctly calculated from the actual spatial mass distribution, where inhomogeneities, as well as the potential due to the mass outside of the core, also contribute to the gravitational energy. We conclude that the observed energy budget of cores in recent surveys is consistent with their non-thermal motions being driven by their self-gravity and in the process of dynamical collapse.