Aims. We investigate the long-term dynamical evolution of two distinct stellar populations of low-mass stars in globular clusters in order to study whether the energy equipartition process can explain the high number of stars harbouring abundance anomalies seen in globular clusters. Methods. We analyse N-body models by artificially dividing the low- mass stars (m\le0.9 M_\odot) into two populations: a small number of stars (second generation) consistent with an invariant IMF and with low specific energies initially concentrated towards the cluster-centre mimic stars with abundance anomalies. These stars form from the slow winds of fast-rotating massive stars. The main part of low-mass (first generation) stars has the pristine composition of the cluster. We study in detail how the two populations evolve under the influence of two-body relaxation and the tidal forces due to the host galaxy. Results. Stars with low specific energy initially concentrated toward the cluster centre need about two relaxation times to achieve a complete homogenisation throughout the cluster. For realistic globular clusters, the number ratio between the two populations increases only by a factor 2.5 due to the preferential evaporation of the population of outlying first generation stars. We also find that the loss of information on the stellar orbital angular momentum occurs on the same timescale as spatial homogenisation. Conclusions. To reproduce the high number of chemically anomalous stars in globular clusters by preserving an invariant IMF, more efficient mechanisms such as primordial gas expulsion are needed to expel the stars in the outer cluster parts on a short timescale.