In this paper, we introduce Multiresolution Mesh Networks-enhanced MeshGraphNets (MGN-MeshGraphNet) for learning mesh-based dynamical fluid simulations. The novelty of our proposal comes from the ability to capture multiscale structures of fluid dynamics via a learnable coarse-graining mechanism on meshes (i.e. mesh multiresolution), along with long-range dependencies between multiple timesteps and resolutions for robust prediction. Our proposed method has shown competitive numerical results in comparison with other machine learning approaches based on graph neural networks. Given the flexibility of our data-driven approach for building mesh multiresolution, our method has better generalizability for new fluid dynamical simulations outside of the training data while attaining high accuracies on multiple resolutions and computational speedup compared to the existing PDE numerical solvers of Navier--Stokes equation.