We investigate here what happens beyond the onset of motion of a droplet on a wall by the action of an imposed shear flow, accounting for inertial effects and contact-angle hysteresis. A diffuse-interface method is used for this purpose, which alleviates the shear stress singularity at a moving contact line, resulting in an effective slip length. Various flow regimes are investigated, including steadily moving drops, and partial or entire droplet entrainment. In the regime of quasi-steadily moving drops, the drop speed is found to be linear in the imposed shear rate, but to exhibit an apparent discontinuity at the onset of motion. The results also include the relation between a local maximum angle between the interface and the wall and the instantaneous value of the contact-line speed. The critical conditions for the onset of entrainment are determined for pinned as well as for moving drops. The corresponding critical capillary numbers are found to be in a rather narrow range, even for quite substantial values of a Reynolds number. The approach to breakup is then investigated in detail, including the growth of a ligament on a drop, and the reduction of the radius of a pinching neck. A model based on an energy argument is proposed to explain the results for the rate of elongation of ligaments. The paper concludes with an investigation of detachment of a hydrophobic droplet from the solid wall.