Flow Matching (FM) is a simulation-free method for learning a continuous and invertible flow to interpolate between two distributions, and in particular to generate data from noise. Inspired by the variational nature of the diffusion process as a gradient flow, we introduce a stepwise FM model called Local Flow Matching (LFM), which consecutively learns a sequence of FM sub-models, each matching a diffusion process up to the time of the step size in the data-to-noise direction. In each step, the two distributions to be interpolated by the sub-flow model are closer to each other than data vs. noise, and this enables the use of smaller models with faster training. This variational perspective also allows us to theoretically prove a generation guarantee of the proposed flow model in terms of the $χ^2$-divergence between the generated and true data distributions, utilizing the contraction property of the diffusion process. In practice, the stepwise structure of LFM is natural to be distilled and different distillation techniques can be adopted to speed up generation. We empirically demonstrate improved training efficiency and competitive generative performance of LFM compared to FM on the unconditional generation of tabular data and image datasets, and also on the conditional generation of robotic manipulation policies.