We present experimental investigations of the spatial and temporal evolution of particle migration in pressure driven flows of Brownian particle suspensions. Binary suspensions of 1.4 micron- and 3.0 micron-diameter colloidal particles are pumped through a 50x500 micron^2 rectangular-cross-section capillary tube. Shear rate gradients caused by the resulting parabolic velocity profile drive particles away from the walls towards the center of the channel where the shear rate is lowest. The flows are directly imaged using high-speed laser scanning confocal microscopy. Size segregation of the particles is observed. Depending on the conditions, either the large or small particles enrich the center. We measure the development of the size segregation by tracking the evolution of the cross-stream concentrations of the particles.