Context. Converging networks of interstellar filaments, that is hubs, have been recently linked to the formation of stellar clusters and massive stars. Understanding the relationship between the evolution of these systems and the formation of cores and stars inside them is at the heart of current star formation research. Aims. The goal is to study the kinematic and density structure of the SDC13 prototypical hub at high angular resolution to determine what drives its evolution and fragmentation. Methods. We have mapped SDC13, a ~1000 M_⊙ infrared dark hub, in NH₃(1,1) and NH₃(2,2) emission lines, with both the Jansky Very Large Array and Green Bank Telescope. The high angular resolution achieved in the combined dataset allowed us to probe scales down to 0.07 pc. After fitting the ammonia lines, we computed the integrated intensities, centroid velocities and line widths, along with gas temperatures and H₂ column densities. Results. The mass-per-unit-lengths of all four hub filaments are thermally super-critical, consistent with the presence of tens of gravitationally bound cores identified along them. These cores exhibit a regular separation of ~0.37 ± 0.16 pc suggesting gravitational instabilities running along these super-critical filaments are responsible for their fragmentation. The observed local increase of the dense gas velocity dispersion towards starless cores is believed to be a consequence of such fragmentation process. Using energy conservation arguments, we estimate that the gravitational to kinetic energy conversion efficiency in the SDC13 cores is ~35%. We see velocity gradient peaks towards ~63% of cores as expected during the early stages of filament fragmentation. Another clear observational signature is the presence of the most massive cores at the filaments’ junction, where the velocity dispersion is largest. We interpret this as the result of the hub morphology generating the largest acceleration gradients near the hub centre. Conclusions. We propose a scenario for the evolution of the SDC13 hub in which filaments first form as post-shock structures in a supersonic turbulent flow. As a result of the turbulent energy dissipation in the shock, the dense gas within the filaments is initially mostly sub-sonic. Then gravity takes over and starts shaping the evolution of the hub, both fragmenting filaments and pulling the gas towards the centre of the gravitational well. By doing so, gravitational energy is converted into kinetic energy in both local (cores) and global (hub centre) potential well minima. Furthermore, the generation of larger gravitational acceleration gradients at the filament junctions promotes the formation of more massive cores.