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 underscores the need for new methods capable of realigning cells. Optimal transport (OT) has emerged as a potent solution, but traditional discrete solvers are hampered by scalability, privacy, and out-of-sample estimation issues. These challenges have spurred the development of neural network-based solvers that parameterize OT maps, known as neural OT solvers. Yet, these models often lack the flexibility needed for broader life sciences applications. To address these deficiencies, our approach introduces stochastic transport plans, relaxes mass conservation rules, and adopts adaptable cost functions, along with integrating quadratic solvers to tackle the complex challenges posed by Gromov-Wasserstein and Fused Gromov-Wasserstein problems. Utilizing the generative power of flow matching, 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.