We present the results of a suite of \Nbody simulations aimed at understanding the fundamental aspects of the long-term evolution of the internal kinematics of multiple stellar populations in globular clusters. Our models enable us to study the cooperative effects of internal, relaxation-driven processes and external, tidally-induced perturbations on the structural and kinematic properties of multiple-population globular clusters. To analyse the dynamical behaviour of the multiple stellar populations in a variety of spin-orbit coupling conditions, we have considered three reference cases in which the tidally perturbed star cluster rotates along an axis oriented in different directions with respect to the orbital angular momentum vector. We focus specifically on the characterisation of the evolution of the degree of differential rotation and anisotropy in the velocity space, and we quantify the process of spatial and kinematic mixing of the two populations. In light of recent and forthcoming explorations of the internal kinematics of this class of stellar systems by means of line-of sight and astrometric measurements, we also investigate the implications of projection effects and spatial distribution of the stars adopted as tracers. The kinematic and structural richness emerging from our models further emphasises the need and the importance of observational studies aimed at building a complete kinematical picture of the multiple population phenomenon.