The field of photorealistic 3D avatar reconstruction and generation has garnered significant attention in recent years; however, animating such avatars remains challenging. Recent advances in diffusion models have notably enhanced the capabilities of generative models in 2D animation. In this work, we directly utilize these models within the 3D domain to achieve controllable and high-fidelity 4D facial animation. By integrating the strengths of diffusion processes and geometric deep learning, we employ Graph Neural Networks (GNNs) as denoising diffusion models in a novel approach, formulating the diffusion process directly on the mesh space and enabling the generation of 3D facial expressions. This facilitates the generation of facial deformations through a mesh-diffusion-based model. Additionally, to ensure temporal coherence in our animations, we propose a consistent noise sampling method. Under a series of both quantitative and qualitative experiments, we showcase that the proposed method outperforms prior work in 4D expression synthesis by generating high-fidelity extreme expressions. Furthermore, we applied our method to textured 4D facial expression generation, implementing a straightforward extension that involves training on a large-scale textured 4D facial expression database.