Mesh-based simulation using Graph Neural Networks (GNNs) has been recognized as a promising approach for modeling fluid dynamics. However, the mesh refinement techniques which allocate finer resolution to regions with steep gradients can induce the over-squashing problem in mesh-based GNNs, which prevents the capture of long-range physical interactions. Conventional graph rewiring methods attempt to alleviate this issue by adding new edges, but they typically complete all rewiring operations before applying them to the GNN. These approaches are physically unrealistic, as they assume instantaneous interactions between distant nodes and disregard the distance information between particles. To address these limitations, we propose a novel framework, called Adaptive Graph Rewiring in Mesh-Based Graph Neural Networks (AdaMeshNet), that introduces an adaptive rewiring process into the message-passing procedure to model the gradual propagation of physical interactions. Our method computes a rewiring delay score for bottleneck nodes in the mesh graph, based on the shortest-path distance and the velocity difference. Using this score, it dynamically selects the message-passing layer at which new edges are rewired, which can lead to adaptive rewiring in a mesh graph. Extensive experiments on mesh-based fluid simulations demonstrate that AdaMeshNet outperforms conventional rewiring methods, effectively modeling the sequential nature of physical interactions and enabling more accurate predictions.