I discuss dynamic properties of multispectral speckle in the context of digital holographic interferometry and image correlation. I outline the correlation of speckles in free space, in an imaging system, and, in the case of interferometric detection, caused by reflection off an inclined diffuse surface. It is shown that interferometric phase gradients and speckle movements are closely related where in fact the phase gradients are the generator of speckle movements in a defocused plane. The theory is exemplified by three typical situations encountered in image-plane digital holographic interferometry.