The blooming diffusion probabilistic models (DPMs) have garnered significant interest due to their impressive performance and the elegant inspiration they draw from physics. Various forms of models have been designed rapidly to augment efficiency and capabilities. Nevertheless, the current theoretical explanations lag behind the advancement of these algorithms, impeding the further progress. In response, we propose FreeFlow, a comprehensive framework that offers a thorough interpretation of the existing DPMs. In FreeFlow, the diffusion process is considered as the time-dependent optimal transport, in which the evolutionary pattern of probability density is determined by the gradient flows of a functional defined in the Wasserstein space. Crucially, our framework necessitates a fluid-mechanics description that not only clarifies the subtle mechanism of DPMs but also indicates the roots of some defects through creative involvement of Lagrangian and Eulerian views. FreeFlow presents a novel perspective governing the evolution of probability through a unified equation. The Riemannian geometry employed in our work has the potential to bridge broader subjects in mathematics, which also enable the involvement of more profound tools for the establishment of more outstanding and generalized models in the future.