Inferring and forecasting latent stochastic dynamics (trajectories) from snapshot data (measurements at different time points across a population, but only one time point per subject) is a critical challenge in areas like single-cell biology. Existing deep learning methods, often based on Schrödinger bridges (SBs), are limited: they either interpolate between only two time points, failing to capture long-term dependencies, or require a pre-set, fixed reference dynamic. We introduce two novel frameworks to overcome these limitations. The first method successfully learns trajectories from multiple time points using only a family of reference dynamics, enhancing trajectory reconstruction. The second, SnapMMD, uses a maximum mean discrepancy (MMD) loss to directly fit the joint state-time distribution. Crucially, SnapMMD allows us to infer unknown and state-dependent volatilities from the data, which is essential for accurate forecasting beyond the observed time horizon. We demonstrate that both approaches significantly improve upon state-of-the-art methods in trajectory inference, velocity-field reconstruction, and forecasting across a variety of real and synthetic experiments.