In recent years, various methods have been proposed for mesh analysis, each offering distinct advantages and often excelling on different object classes. We present a novel Mixture of Experts (MoE) framework designed to harness the complementary strengths of these diverse approaches. We propose a new gate architecture that encourages each expert to specialise in the classes it excels in. Our design is guided by two key ideas: (1) random walks over the mesh surface effectively capture the regions that individual experts attend to, and (2) an attention mechanism that enables the gate to focus on the areas most informative for each expert's decision-making. To further enhance performance, we introduce a dynamic loss balancing scheme that adjusts a trade-off between diversity and similarity losses throughout the training, where diversity prompts expert specialization, and similarity enables knowledge sharing among the experts. Our framework achieves state-of-the-art results in mesh classification, retrieval, and semantic segmentation tasks. Our code is available at: https://github.com/amirbelder/MME-Mixture-of-Mesh-Experts.