Flow matching has emerged as a powerful alternative to diffusion models for generative modeling, but standard conditional flow matching is fundamentally misaligned with partially observed time series: starting from an uninformed Gaussian source forces the model to transport even already-observed entries from noise to the target, wasting capacity on trivially known regions. We propose MissPath-FM, a partial-observation-aware flow matching framework that redesigns the source prior and probability path for incomplete time series. MissPath-FM employs structured source priors (interpolation priors and Gaussian process posterior priors) to initialize the flow close to the observed signal, paired with a missingness-aware smooth path that allocates transport asymmetrically between observed and missing regions. We evaluate systematically across four datasets, four missingness mechanisms (MCAR, block, MNAR, irregular), two missing ratios, and three sequence lengths (T in {24, 48, 96}), totaling 48 settings against eight baselines. MissPath-FM wins 29 of 32 settings at T=48 and 45 of 48 overall (93.8%). Our ablations reveal a regime-dependent design hierarchy in which no single component suffices: under random missingness the source prior yields a large initial gain, but under structured block missingness it can fail entirely and the learned flow becomes indispensable; the missingness-aware path contributes a further 12%-26% across all regimes. Mechanistically, structured priors shorten observed-region transport by up to 15x, concentrating the velocity field on genuinely uncertain entries. Project page: https://zgp110.github.io/misspath-fm/

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