This paper addresses the phenomenon of random packing of sedimenting grains in the context of metallic alloy solidification from a dynamics perspective. More precisely, we investigate the evolution of the grains from the initial steady-state sedimentation until the final mechanical equilibrium-packing-is achieved. The packing dynamics of two grain geometries (spherical and nonconvex) is investigated for a hydrodynamic condition of a low ratio of grain inertia to viscous dissipation, given by a low Stokes number, St, of o(10^{-3}), which is typical in solidification. An experimental setup that consists of a vertical column filled with glycerol is employed to reproduce the packing of the sedimenting grains by means of two different protocols (sequential and collective), enabling to track the trajectory of the grains. The grain packing is also simulated by means of a discrete element method for both grain geometries in case of collective packing protocol, complementing the experimental work-trajectory tracking-with the time evolution of the local solid fraction, of the local number of contacting neighbors and of the grain orientation during the packing dynamics.