The origin of multiple stellar populations in Globular Clusters (GCs) is one of the greatest mysteries of modern stellar astrophysics. N-body simulations suggest that the present-day dynamics of GC stars can constrain the events that occurred at high redshift and led to the formation of multiple populations. Here, we combine multi-band photometry from the Hubble Space Telescope (HST) and ground-based facilities with HST and Gaia Data Release 2 proper motions to investigate the spatial distributions and the motions in the plane of the sky of multiple populations in the type II GCs NGC 5139 (Omega Centauri) and NGC 6656 (M 22). We first analyzed stellar populations with different metallicities. Fe-poor and Fe-rich stars in M 22 share similar spatial distributions and rotation patterns and exhibit similar isotropic motions. Similarly, the two main populations with different iron abundance in Omega Centauri share similar ellipticities and rotation patterns. When analyzing different radial regions, we find that the rotation amplitude decreases from the center towards the external regions. Fe-poor and Fe-rich stars of Omega Centauri are radially anisotropic in the central region and show similar degrees of anisotropy. We also investigate the stellar populations with different light-element abundances and find that their N-rich stars exhibit higher ellipticity than N-poor stars. In Omega Centauri Centauri both stellar groups are radially anisotropic. Interestingly, N-rich, Fe-rich stars exhibit different rotation patterns than N-poor stars with similar metallicities. The stellar populations with different nitrogen of M 22 exhibit similar rotation patterns and isotropic motions. We discuss these findings in the context of the formation of multiple populations.