The faster-moving particles exhibit higher kinetic energy. The flow of energy takes place from high to low, thus, uniformizing the distribution of energy between the particles. So, the slower-moving particles end up moving a bit faster, and the faster-moving particles end up moving a bit slower.
Presuming that they are ideal particles, which conserve kinetic energy, they would all approach the similar kinetic energy, and thus the same temperature and root-mean-squared root.
