We have observed P-Cygni and asymmetric, blue-shifted absorption profiles in the He I 10830 lines of twelve R Coronae Borealis (RCB) stars over short (1 month) and long (3 year) timescales to look for variations linked to their dust-formation episodes. In almost all cases, the strengths and terminal velocities of the line vary significantly and are correlated with dust formation events. Strong absorption features with blue-shifted velocities ~400 km/s appear during declines in visible brightness and persist for about 100 days after recovery to maximum brightness. Small residual winds of somewhat lower velocity are present outside of the decline and recovery periods. The correlations support models in which recently formed dust near the star is propelled outward at high speed by radiation pressure and drags the gas along with it.