Supported lipid bilayers (SLBs) are one of the most common model systems for cell membrane studies. We have previously found that when applying a bulk flow of liquid above an SLB the lipid bilayer and its constituents move in the direction of the bulk flow in a rolling type of motion, with the lower monolayer being essentially stationary. In this study, a theoretical platform is developed to model the dynamic behavior of a shear-driven SLB. In most regions of the moving SLB, the dynamics of the lipid bilayer is well explained by a balance between the hydrodynamic shear force arising from the bulk flow above the lipid bilayer and the friction between the upper and lower monolayers of the SLB. These two forces result in a drift velocity profile for the lipids in the upper monolayer of the SLB that is highest at the center of the channel and decreases to almost zero at the corners of the channel. However, near the front of an advancing SLB a very different flow behavior is observed, showing an almost constant drift velocity of the lipids over the entire bilayer front. In this region, the motion of the SLB is significantly influenced by gradients in the surface pressure as well as internal friction due to molecules that have accumulated at the front of the SLB. It is shown that even a modest surface fraction of accumulated molecules (similar to 1%) can drastically affect the behavior of the SLB near the bilayer front, forcing the advancing lipids in the SLB away from the center of the channel out toward the sides.