ABSTRACT We study the linear growth and non-linear saturation of the ‘acoustic Resonant Drag Instability’ (RDI) when the dust grains, which drive the instability, have a wide, continuous spectrum of different sizes. This physics is generally applicable to dusty winds driven by radiation pressure, such as occurs around red-giant stars, star-forming regions, or active galactic nuclei. Depending on the physical size of the grains compared to the wavelength of the radiation field that drives the wind, two qualitatively different regimes emerge. In the case of grains that are larger than the radiation’s wavelength – termed the constant-drift regime – the grain’s equilibrium drift velocity through the gas is approximately independent of grain size, leading to strong correlations between differently sized grains that persist well into the saturated non-linear turbulence. For grains that are smaller than the radiation’s wavelength – termed the non-constant-drift regime – the linear instability grows more slowly than the single-grain-size RDI and only the larger grains exhibit RDI-like behaviour in the saturated state. A detailed study of grain clumping and grain–grain collisions shows that outflows in the constant-drift regime may be effective sites for grain growth through collisions, with large collision rates but low collision velocities.