This study presents a comprehensive investigation of U-Net-based graph neural networks (Graph U-Nets) for mesh-agnostic spatio-temporal forecasting of unsteady flow fields. We systematically adapt and enhance Graph U-Nets, originally developed for classification tasks, for high-dimensional regression problems in fluid dynamics through extensive architectural modifications and hyperparameter optimization. Key enhancements include the implementation of Gaussian mixture model convolutional operators, which provide superior flexibility in modeling node dynamics and reduce prediction error by 95% compared to conventional graph convolutional operators. Additionally, we introduce noise injection strategies that significantly improve long-term prediction robustness, achieving an 86% reduction in temporal prediction error. Through comprehensive ablation studies, we investigate the effects of pooling strategies, normalization techniques, and architectural choices on model performance. We demonstrate the framework’s effectiveness in both transductive learning settings—successfully predicting flow fields in unseen spatial regions—and inductive learning scenarios across diverse mesh configurations with varying vortex shedding dynamics. Notably, we discover that optimal inductive performance requires eliminating pooling operations and employing layer normalization, contrary to single-mesh scenarios. The enhanced Graph U-Net successfully generalizes to unseen mesh scenarios, achieving improved prediction accuracy for challenging slow-vortex-shedding cases when trained on diverse flow regimes. This work establishes Graph U-Nets as a viable and flexible alternative to convolutional neural networks for computational fluid dynamics applications, demonstrating their potential for real-time flow prediction in digital twin frameworks across diverse industrial scenarios.