Multistable morphing structures have gained considerable attention over the past three decades due to their ability to efficiently work in various operating conditions. Unsymmetrical laminates, which yields two stable equilibrium shapes due to the residual thermal stresses, are widely investigated to produce multistable structures. Even though cured shapes of unsymmetrical laminates are promising candidates for the design of morphing structural components, they may not fulfil all the requirements of a continuous morphing structure requiring more than two stable states during the morphing action. As a solution to achieve more than two stable configurations, series-connected unsymmetrical laminates are proposed in the present analysis. Connecting different bistable laminates results in multiple stable configurations at the end of the curing process, which is often desired in continuous shape-changing applications. The snap-through process involving shape transition between the generated stable shapes is highly nonlinear in nature. Since morphing aerospace structures are often subjected to severe dynamic excitation, large-amplitude nonlinear vibrations are inevitable during the snap-through transition. This work aims to explore the dynamic characteristic of multistable continuous composite plates generated by connecting bistable laminates without any external fixing aids. The proposed numerical analysis is carried out within a commercially available finite element package, ABAQUS. The effect of aspect ratios, laminate layups and actuation loads on the dynamic characteristic (natural frequencies and mode shapes) are reported from systematic parametric studies. The proposed model is further extended to predict the nonlinear characteristics of single well vibration and cross well vibration.