Single-cell genomics has significantly advanced our understanding of cellular behavior, catalyzing innovations in treatments and precision medicine. However,single-cell sequencing technologies are inherently destructive and can only measure a limited array of data modalities simultaneously. This limitation underscoresthe need for new methods capable of realigning cells. Optimal transport (OT)has emerged as a potent solution, but traditional discrete solvers are hampered byscalability, privacy, and out-of-sample estimation issues. These challenges havespurred the development of neural network-based solvers, known as neural OTsolvers, that parameterize OT maps. Yet, these models often lack the flexibilityneeded for broader life science applications. To address these deficiencies, ourapproach learns stochastic maps (i.e. transport plans), allows for any cost function,relaxes mass conservation constraints and integrates quadratic solvers to tackle thecomplex challenges posed by the (Fused) Gromov-Wasserstein problem. Utilizingflow matching as a backbone, our method offers a flexible and effective framework.We demonstrate its versatility and robustness through applications in cell development studies, cellular drug response modeling, and cross-modality cell translation,illustrating significant potential for enhancing therapeutic strategies.