Multi-stable composites are a novel type of composites capable of adopting multiple statically stable configurations. Due to the multi-stability property this type of composite material has been considered for several applications, particularly for morphing structures. The change of shape between stable states is achieved by a nonlinear mechanism known as snap-through. Most of the research done on these composites has focused on predicting the configuration after manufacture, its static characteristics and static actuation strategies to induce snap-through. However, these structures will operate subject to dynamic loads. Yet, very little work has been carried out to examine the dynamic behaviour of bi-stable composites. This paper focuses on the study of the cross-well dynamics of a bi-stable composite plate. A simple model previously derived for the dynamics confined to a single stable state is extended to include cross-well dynamics. The rich dynamics are experimentally investigated, focusing on cross-well oscillations and the key dynamic features of snap-through. Numerical simulations are obtained and compared to the experimental results showing good agreement. In particular, experimentally observed characteristics suggesting chaotic oscillations for cross-well dynamics are captured well by the proposed model. The results herein could be used for implementing control strategies for both configuration morphing and undesired snap-through suppression of bi-stable composites.