We study the dynamics of superconducting vortices in Nb rings as the s
ystem is continuously driven to the depinning threshold by the slow ra
mping of an external magnetic field. Miniature Hall probes simultaneou
sly detect local and global flux changes arising from vortex motion. W
ith decreasing temperature, the dynamics evolve continuously from smoo
th flow to several types of avalanche behavior. In particular, we obse
rve a crossover from broad to narrow size distributions of avalanche e
vents which correspond to global decreases in the flux density gradien
t rather than local redistributions of flux. We show how this evolutio
n can arise from the magnetothermal instability of the Bean state.