Experiments are presented elucidating how the evolution of foam microstructure by gas diffusion from high to low pressure bubbles can significantly speed up the rate of gravitational drainage, and vice versa. This includes detailed data on the liquid-fraction dependence of the coarsening rate, and on the liquid-fraction and the bubble-size profiles across a sample. These results can be described by a "coarsening equation" for the increase of bubble growth rate for drier foams. Spatial variation of the average bubble size and liquid fraction can also affect the growth and drainage rates.