This paper extends our recent study on Casimir friction forces for dielectric plates moving parallel to each other (Høye and Brevik 2014 Eur. Phys. J. D 68 61), to a case where the plates are no longer restricted to rectilinear motion. Part of the mathematical formalism thereby becomes more cumbersome, but reduces in the end to the form that we expected to be the natural one in advance. As an example, we calculate the Casimir torque on a planar disc rotating with constant angular velocity around its vertical symmetry axis next to another plate.