Many real-world robotic applications require dynamically sensitive manipulation, where success depends not only on reaching a target state but on maintaining stable object dynamics throughout execution. We study the stable transport of liquid-filled containers, where a robot must move objects to target locations while suppressing sloshing and preventing spillage. Unlike conventional pick-and-place, this task imposes stringent requirements on motion smoothness and trajectory-level stability, exposing clear limitations in existing systems. Specifically, fluid simulation remains too costly for online reinforcement learning; human teleoperation introduces unintended accelerations that induce sloshing during imitation learning; and current policy pipelines optimize for task completion rather than dynamic stability. We propose a synthetic-to-real framework coupling physically validated data generation with a hierarchical, diffusion-based controller. The scalable data pipeline synthesizes grasps, filters unstable poses via a vision-language model, and validates transport trajectories through fluid simulation. The policy is organized with a high-level module that translates language and visual observations into SE(3) control targets, and a latent diffusion controller that first plans efficiently in a compact latent space and then decodes dense action chunks, enabling the high control frequency needed for smooth and stable motion. Extensive experiments show our system outperforms state-of-the-art manipulation policies in transport smoothness and dynamic stability.