We study the dynamics of superconducting vortices in Nb rings as the system is continuously driven to the depinning threshold by the slow ramping of an external magnetic field. Miniature Hall probes simultaneously detect local and global flux changes arising from vortex motion. With decreasing temperature, the dynamics evolve continuously from smooth flow to several types of avalanche behavior. In particular, we observe a crossover from broad to narrow size distributions of avalanche events which correspond to global decreases in the flux density gradient rather than local redistributions of flux. We show how this evolution can arise from the magnetothermal instability of the Bean state.