The scattering of a spherical electromagnetic wave from a radially moving spherical discontinuity in index of refraction is analyzed. The Doppler shifts are found to depend on the radius as well as the velocity of the discontinuity. As an example the scattering from a tenuous evaporating dielectric sphere is found. The plane‐wave scattering problem is formulated and it is shown that the scattered frequencies have a satisfying physical interpretation in the geometrical optics limit.