Many multicomponent materials exhibit significant configurational disorder. Diffusing ions in such materials migrate along a network of sites that have different energies and that are separated by configuration dependent activation barriers. We describe a formalism that enables a first-principles calculation of the diffusion coefficient in solids exhibiting configurational disorder. The formalism involves the implementation of a local cluster expansion to describe the configuration dependence of activation barriers. The local cluster expansion serves as a link between accurate first-principles calculations of the activation barriers and kinetic Monte Carlo simulations. By introducing a kinetically resolved activation barrier, we show that a cluster expansion for the thermodynamics of ionic disorder can be combined with a local cluster expansion to obtain the activation barrier for migration in any configuration. This ensures that in kinetic Monte Carlo simulations, detailed balance is maintained at all times and kinetic quantities can be calculated in a properly equilibrated thermodynamic state. As an example, we apply this formalism for an investigation of lithium diffusion in ${\mathrm{Li}}_{x}{\mathrm{CoO}}_{2}.$ A study of the activation barriers in ${\mathrm{Li}}_{x}{\mathrm{CoO}}_{x}$ within the local density approximation shows that the migration mechanism and activation barriers depend strongly on the local lithium-vacancy arrangement around the migrating lithium ion. By parametrizing the activation barriers with a local cluster expansion and applying it in kinetic Monte Carlo simulations, we predict that lithium diffusion in layered ${\mathrm{Li}}_{x}{\mathrm{CoO}}_{2}$ is mediated by divacancies at all lithium concentrations. Furthermore, due to a strong concentration dependence of the activation barrier, the predicted diffusion coefficient varies by several orders of magnitude with lithium concentration x.