We demonstrate experimentally and theoretically that the dissipative state of superconducting samples with a periodic array of holes at high current densities consists of flux rivers resulting from a short-range attractive interaction between vortices. This dynamically induced vortex-vortex attraction results from the migration of quasiparticles out of the vortex core (kinematic vortices). We have directly visualized the formation of vortex chains by scanning Hall probe microscopy after freezing the dynamic state by a field cooling procedure at a constant bias current. Similar experiments carried out in a sample without holes show no hint of flux river formation. We shed light on this nonequilibrium phenomena modeled by time-dependent Ginzburg-Landau simulations.