A pulsating thermal-diffusive instability is predicted at all non-unity Lewis numbers for a simple configuration representing planar diffusional burning of a cloud of fuel droplets moving through gas. The mechanism driving the instability is the coupling of the heat source at the flame sheet to the vapor source at the cloud perimeter. For hydrocarbon fuels, pulsations are predicted for oxygen concentrations well below ambient. This result indicates that thermal-diffusive instabilities can occur in the flame-sheet limit, suggesting new opportunities for investigating the largely unexplored field of the thermal-diffusive dynamics of diffusion flames.