The deformation of the electric double layer around a charged colloidal particle during sedimentation or electrophoresis in a binary, symmetric electrolyte is studied. The surface potential of the particle is assumed to be small compared to the thermal voltage scale. Additionally, the Debye length is assumed to be large compared to the particle size. These assumptions enable a linearization of the electrokinetic equations. The particle appears as a point charge in this thick-double-layer limit; the distribution of charge in the diffuse cloud surrounding it is determined by a balance of advection due to the particle motion, Brownian diffusion of ions, and electrostatic screening of the particle by the cloud. The ability of advection to deform the charge cloud from its equilibrium state is parametrized by a Peclet number, Pe. For weak advection (Pe << 1), the cloud is only slightly deformed. In contrast, the cloud can be completely stripped from the particle at Pe >> 1; consequently, electrokinetic effects on the particle motion vanish in this regime. Therefore, in sedimentation the drag limits to Stokes' law for an uncharged particle as Pe -> infinity. Likewise, the particle velocity for electrophoresis approaches Huckel's result. The strongly deformed cloud at large Pe is predicted to generate a concomitant increase in the sedimentation field in a dilute settling suspension.