The measurement of phase in coherent electron systems--that is, 'mesoscopic' systems such as quantum dots--can yield information about fundamental transport properties that is not readily apparent from conductance measurements. Phase measurements on relatively large quantum dots recently revealed that the phase evolution for electrons traversing the dots exhibits a 'universal' behaviour, independent of dot size, shape, and electron occupancy. Specifically, for quantum dots in the Coulomb blockade regime, the transmission phase increases monotonically by pi throughout each conductance peak; in the conductance valleys, the phase returns sharply to its starting value. The expected mesoscopic features in the phase evolution--related to the dot's shape, spin degeneracy or to exchange effects--have not been observed, and there is at present no satisfactory explanation for the observed universality in phase behaviour. Here we report the results of phase measurements on a series of small quantum dots, having occupancies of between only 1-20 electrons, where the phase behaviour for electron transmission should in principle be easier to interpret. In contrast to the universal behaviour observed thus far only in the larger dots, we see clear mesoscopic features in the phase measurements when the dot occupancy is less than approximately 10 electrons. As the occupancy increases, the manner of phase evolution changes and universal behaviour is recovered for some 14 electrons or more. The identification of a transition from the expected mesoscopic behaviour to universal phase evolution should help to direct and constrain theoretical models for the latter.