Lateral variations in the thermal boundary conditions at the horizontal boundary of an otherwise stably stratified fluid layer drive circulatory motion. For a rapidly rotating electrically conducting fluid, the introduction of a background vertical magnetic field reduces the dimensionless strength of the thermal wind from $O(1)$ to $O(E^{1/4})$ for prescribed temperature variation or to $O(E^{1/2})$ for prescribed heat-flux variation, where $E$ is the Ekman number. This is a significant effect when considering the spatially variable cooling of the Earth's core by the mantle. A general discussion of linear hydromagnetic flows identifies a large number of lengthscales inherent in the differential system as functions of the Ekman, Elsasser and stratification numbers, and shows that other scalings arise from the boundary conditions.