The electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity with an arbitrary thickness of the electric double layers adjacent to the particle and cavity surfaces is analyzed at the quasisteady state When the zeta potentials associated with the solid surfaces are arbitrarily nonuniform. Through the use of the multipole expansions of the zeta potentials and the linearized Poisson-Boltzmann equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately solved. The modified Stokes equations governing the fluid velocity field are dealt with using a generalized reciprocal theorem, and explicit formulas for the electrophoretic and angular velocities of the particle valid for all values of the particle-to-cavity size ratio are obtained. To apply these formulas, one only has to calculate the monopole, dipole, and quadrupole moments of the zeta potential distributions at the particle and cavity surfaces. In some limiting cases, our result reduces to the analytical solutions available in the literature. In general, the boundary effect on the electrophoretic motion of the particle is a qualitatively and quantitatively sensible function of the thickness of the electric double layers relative to the radius of the cavity.