Biomicrofluidics research such as single-cell analysis, cell-sorting, and droplet-based microchemical reactors often involves isolating a single cell or drop in a bioreactor chamber and exposing it to controlled flow of nutrient-rich solution. We use three-dimensional moving-frame boundary-integral simulations along with macroscopic flow-cell experiments to study the motion of a deformable droplet under gravity into and out of a well with nearly sharp corners in the Stokes flow regime. The dynamics of the droplet is influenced by the drop-to-bulk fluid viscosity ratio, capillary number, Archimedes number, and the chamber length. Increasing the Archimedes number (ratio of gravity and viscous forces) causes the drop to settle towards the bottom of the chamber, while a smaller Archimedes number allows the drop to escape without any hook or tail. An intermediate Archimedes number may lead to droplet breakup, with the parent drop either becoming trapped or escaping after forming a thin neck near the corner. Higher capillary numbers lead to large, worm-like drop elongation, with the tail becoming trapped and the head elongating out of the channel.