This work introduces a novel dynamical reduced-order approximation method for partial differential equations (PDEs) that exploits the geometric structure of the solution manifold to construct an adaptive reduced representation. The proposed approach evolves the parametrization of the reduced basis through suitably constructed systems of ordinary differential equations, enabling the approximation space to adapt dynamically. The evolution of the approximation space is shown to be optimal with respect to the metric induced in the tangent space of the solution manifold, thereby ensuring an efficient and accurate representation of the underlying dynamics. The method is illustrated in the context of Wasserstein gradient flows, which arise in a wide range of applications spanning advection-diffusion-reaction PDEs and Optimization, and whose numerical treatment becomes particularly challenging in high-dimensional settings. In this regime, the numerical approximation of Wasserstein gradient flows is challenged by the curse of dimensionality, the prohibitive cost of optimal transport computations, and sampling inefficiencies in the representation of evolving probability measures. These difficulties are compounded by nonlinear interactions and the simultaneous presence of diffusive and transport-dominated dynamics. In contrast, the proposed dynamical approach does not require a mesh, attains high accuracy with a small number of basis functions, and requires only a single sampling of the initial condition. The proposed framework contributes to the development of a general nonlinear reduced-order model (ROM) capable of accurately describing a wide range of physical dynamics. Its effectiveness is demonstrated on well-known Wasserstein gradient flows, including linear and double-well Fokker–Planck equations, the Porous Medium equation, and related models.