Morphing solutions of partial differential equations is a fundamental issue in model reduction to overcome the so-called Kolmogorov barrier. This fundamental limit relates to the actual possibility of describing a solution manifold of parametric PDEs by a linear or affine representation for a given subspace dimension. One way to circumvent this limit is by learning bijective mappings based on modeling the evolution of appropriate markers of the solution. Gradient flows in the Wasserstein metric describe the evolution of probability measures by following the steepest descent of an energy functional with respect to the Wasserstein distance. These flows can be linked to the Fokker-Planck (FP) equation \cite{Jordan1998}, which describes the time evolution of probability distributions under the influence of drift and diffusion processes. We propose using gradient flows inspired by the FP equation to determine suitable bijective mappings of possibly non-simply connected domains with curved boundaries, in a fast and rigorous way, extending the works \cite{iollo2022mapping} and \cite{cucchiara2024model}. \begin{thebibliography}{99} \bibitem{Jordan1998} R. Jordan, D. Kinderlehrer, and F. Otto, \textit{The Variational Formulation of the Fokker–Planck Equation}, SIAM Journal on Mathematical Analysis, 29 (1), 1-17, 1998. \bibitem{iollo2022mapping} A. Iollo, A. and T. Taddei. \textit{Mapping of coherent structures in parameterized flows by learning optimal transportation with Gaussian models}, Journal of Computational Physics, 471, 111671, 2022. \bibitem{cucchiara2024model} S. Cucchiara, A. Iollo, T. Taddei, and H. Telib. \textit{Model order reduction by convex displacement interpolation}. Journal of Computational Physics, 514, 113230, 2024. \end{thebibliography}