Collisions can change individual molecular speeds but the distribution of speeds remains the same.
At the same temperature, lighter gases move on average faster than heavier gases. At higher temperatures at greater fraction of the molecules are moving at higher speeds. In general the mean, the root mean square and the most probable value in a distribution are all different. Below, the blue line is a cold gas and the red line is a hot gas. Note that the rms speed, u as well as the entire speed distribution changes with temperature for a given gas.