A general formalism is developed to examine the sedimentation of a charged particle in a bounded system at small Peclet numbers and small particle-surface potentials. The excess viscous force is evaluated using a generalized form of the Lorentz reciprocal theorem, eliminating the need to calculate the detailed fluid flow around the charged particle. Specific calculations are provided for the sedimentation of a charged sphere in a concentric (uncharged) spherical cavity. Boundary interactions increase the magnitude of the excess force at small Debye lengths with the opposite effect seen at very large Debye lengths. A general solution is presented for the sedimentation velocity for art arbitrary particle in both unbounded and bounded systems in the limit of very thin double layers. The excess force under these conditions is proportional to the square of the Debye length, with the proportionality constant being a function of the detailed system geometry. These results provide important insights into the transport of charged particles in porous membranes and chromatographic materials.