The overturn of mantle cumulates following the crystallization of the lunar magma ocean – particularly the sinking of ilmenite-bearing cumulates (IBC) – provides an explanation for several aspects of the Moon’s evolution. However, the growth of a stagnant lid due to the temperature dependence of the viscosity tends to prevent IBC from sinking. Here, we investigate the dynamics of the overturn based on a composition-dependent rheology coupling diffusion and dislocation creep of major lunar mantle minerals via three alternative mixing models: isostrain, isostress and Minimized Power Geometric (MPG). The pre-overturn structure is obtained from a fractional crystallization model of the lunar magma ocean, which predicts the formation of a 36-km-thick IBC layer. The possibility of overturning this layer strongly depends on the choice of the rheological mixing model. The isostress model allows for a rapid and complete overturn, while the isostrain and the experiment-based MPG models do not allow IBC to sink. If IBC started sinking and mixing with the underlying mantle during magma ocean solidification, IBC could be initially distributed across a layer with a thickness of up to 150 km, whose partial overturn is always possible, independent of the rheological model. These results highlight the importance of improving rheological models for relevant lunar materials. In all cases, the overturn occurs via small-scale instabilities.