We have studied the properties of the dynamic partially coherent speckles in the Fresnel and Fraunhofer diffraction regions. We use the theory of propagation of the mutual intensity in partially coherent light, and derive the general form of the spatio-temporal cross-correlation functions of the dynamic speckle intensities produced by a moving diffusing object. Based on these theoretical expressions, the characteristics of the movement and the boiling of the dynamic partially coherent speckles are discussed. By choosing the proper extended incoherent light source and by adjusting the parameters of the optical setup, we have observed experimentally the partially coherent speckle in the Fresnel and Fraunhofer diffraction regions. Then we study qualitatively the movement and the boiling of the speckles. Using techniques of photon counting and automatic data acquisition, we have measured the temporal evolution of the speckle intensity in the Fraunhofer plane as the object moves, and thus measured the cross-correlation of the speckles versus the temporal increment. The experimental data are in agreement with the theoretical results.