Following a novel realization of low-Reynolds-number swimming (Dreyfus et al., Nature, vol. 436, 2005, p. 862), in which self-assembled filaments of paramagnetic micron-sized beads are tethered to red blood cells and then induced to swim under crossed uniform and oscillating magnetic fields, the dynamics of magnetoelastic filaments is studied. The filament is modelled as a slender elastica driven by a magnetic body torque. The model is applied to experiments of Goubault et al. (Phys. Rev. Lett., vol. 91, 2003, art. 260802) to predict the lifetimes of metastable static filament conformations that are known to form under uniform fields. A second experimental swimming scenario, complementary to that of Dreyfus et al. (2005), is described: filaments are capable of swimming even if not tethered to red blood cells. Yet, if both ends of the filament are left free and the material and magnetic parameters are uniform along its length then application of an oscillating transverse field can only generate homogeneous torques, and net translation is prohibited by symmetry. It is shown that fore–aft symmetry is broken when variation of the bending stiffness along the filament is accounted for by including elastic defects, which produces results consistent with the swimming phenomenology.