The paper considers the flow field during spin-up from rest of liquid metal in a cylindrical stationary cavity due to a rotating transverse uniform low-frequency magnetic field. It is assumed that the Ekman and the magnetic Reynolds numbers are small. An approximate model, based on matching of Bödewadt-type layers with an inviscid core, with possible influence from the sidewall, for laminar flow is developed. It is shown that the angular velocity in the core is a function of time only. Analytical solutions for the angular velocity and the meridional flow in the core are presented, and supplemented by finite-difference results to show the sidewall effects. The spin-down following the switch-off of the magnetic forcing, the influence of the axial variations of the magnetic field, and the relevance to turbulent flows are discussed.