In the growth of nanoscale device structures, the ultimate goal is atomic-level precision. By growing III-V nanowires in a transmission electron microscope, we measured the local kinetics in situ as each atomic plane was added at the catalyst-nanowire growth interface by the vapor-liquid-solid process. During growth of gallium phosphide nanowires at typical V/III ratios, we found surprising fluctuations in growth rate, even under steady growth conditions. We correlated these fluctuations with the formation of twin defects in the nanowire, and found that these variations can be suppressed by switching to growth conditions with a low V/III ratio. We derive a growth model showing that this unexpected variation in local growth kinetics reflects the very different supply pathways of the V and III species. The model explains under which conditions the growth rate can be controlled precisely at the atomic level.