An effective diffusivity model is developed for mass transport by molecular diffusion and unsteady solute buoyancy-driven convection in a horizontal capillary by a method similar to that used for Taylor dispersion in forced flow. A ID non-linear diffusion equation is obtained and an analytical asymptotic solution is found for dominant convective mass transport, the solution for dominant molecular diffusion is already well known. Both the effective diffusivity model and the asymptotic solutions are validated numerically. A scaling analysis clarifies the boundary between dominant molecular diffusion and dominant convective mass transport. The use of a steady vertical magnetic field for liquid metals and semi-conductors is found to damp the convective mass transport by a factor of Ha(-4), where Ha is the Hartmann number, characterising the MHD problem.