Sampling from Boltzmann distributions of molecular systems is an inference problem that has seen significant recent developments fuelled by advances in neural density estimation. We develop a novel sampling algorithm, Neural Transport Nested Sampling (NTNS), which combines the classical strengths of nested sampling with modern neural flow-based methods. NTNS uses a flow matching velocity as the drift in a Metropolis–Hastings corrected Langevin kernel inside a nested sampling outer loop, requiring only evaluations of the target energy function and providing scalable estimation of the full partition function of high-dimensional particle systems. We benchmark NTNS on challenging molecular sampling benchmarks, scaling up to Lennard–Jones clusters of 55 interacting particles, where it reduces both interatomic distance and energy Wasserstein errors to reference MCMC by over an order of magnitude relative to the strongest neural baselines at lower wall-clock cost. To our knowledge, NTNS is also the first neural sampler to return a calibrated, temperature resolved partition function estimate at this scale, recovering the phase structure across temperature from a single run.