We consider the dispersion of waves in a rapidly rotating, Boussinesq fluid which is threaded by a magnetic field and stirred by slowly gravitating buoyant blobs. Motivated by dynamics in the core of the Earth, we focus on the evolution of inertial-Alfven wave packets radiated from the buoyant anomalies. These waves resemble conventional low-frequency inertial waves, in the sense that energy disperses on the fast timescale of the background rotation rate, though they also exhibit slower Alfven-like propagation along magnetic field lines. When the magnetic field is uniform, inertial-Alfven waves automatically focus energy radiation onto the rotation axis, a property they share with conventional low-frequency inertial waves in the hydrodynamic case, and which ensures that they dominate the dispersion pattern. However, the situation changes significantly when the magnetic field, B, is non-uniform. In particular, we show any non-uniformity of B causes inertial-Alfven waves to evolve into a more general form of magnetic-Coriolis (MC) wave, and that these waves refract, dispersing somewhat offaxis. Moreover, if inertial-Alfven waves are launched near the equator they can be confined to low latitudes by a critical layer at which the axial group velocity drops to zero. Given that the magnetic field in a planetary core is inevitably non-uniform, we conclude that quasi-geostrophy is most likely achieved through a combination of weakly modified inertial waves and a form of slightly off-axis MC wave in which the inertial and Alfven frequencies are comparable.