We examine the motion and deformation of air bubbles and viscous drops through vertical cylindrical capillaries in the presence of an imposed pressure-driven flow. Experimental measurements of the terminal velocity of drops and bubbles are reported for a wide range of drop sizes in a variety of two-phase systems, and the steady drop shapes are quantitatively characterized using digital image analysis. In contrast to the pressure-driven motion of neutrally-buoyant drops, the relative mobility of a buoyant drop is not a monotonically increasing function of capillary number. The relative mobility is enhanced as the buoyancy force becomes more dominant compared to surface tension, or as the drop fluid becomes less viscous relative to the suspending fluid. However, there is a limiting value of the Bond number beyond which the relative mobility becomes insensitive to the value of the Bond number. Similarly, the thickness of the liquid film surrounding large drops increases rapidly with increasing Bond number, but eventually approaches a constant value as the Bond number exceeds the limiting value. This limiting value of the Bond number is found to be a decreasing function of capillary number. When buoyancy and pressure forces act in the same direction, increasing the Bond number is found to delay the formation of a re-entrant cavity at the trailing end of the drop.