Numerous observational studies have revealed the ubiquitous presence of multiple stellar populations in globular clusters and cast many hard challenges for the study of the formation and dynamical history of these stellar systems. In this Letter we present the results of a study of the kinematic properties of multiple populations in NGC 2808 based on high-precision Hubble Space Telescope proper-motion measurements. In a recent study, Milone et al. have identified five distinct populations (A, B, C, D, and E) in NGC 2808. Populations D and E coincide with the helium-enhanced populations in the middle and the blue main sequences (mMS and bMS) previously discovered by Piotto et al.; populations A, B, and C correspond to the redder main sequence (rMS) that in the Piotto et al. was associated with the primordial stellar population. Our analysis shows that, in the outermost regions probed (between about 1.5 and 2 times the cluster half-light radius), the velocity distribution of populations D and E is radially anisotropic (the deviation from an isotropic distribution is significant at the ~3.5-sigma level). Stars of populations D and E have a smaller tangential velocity dispersion than those of populations A, B, and C, while no significant differences are found in the radial-velocity dispersion. We present the results of a numerical simulation showing that the observed differences between the kinematics of these stellar populations are consistent with the expected kinematic fingerprint of the diffusion towards the cluster outer regions of stellar populations initially more centrally concentrated. ; Comment: 6 pages, 5 figure, accepted for publication in ApJ Letters