We examine optimal matchings or transport between two stationary random measures. It encompasses allocation from Lebesgue measure to a point process or matching a point process to a regular (shifted) lattice. The main focus of the article is the implication of hyperuniformity (reduced variance fluctuations in point processes) to optimal transport: in dimension 2, we show that the typical matching cost has finite second moment under a mild logarithmic integrability condition on the reduced pair correlation measure, showing that most planar hyperuniform point processes are L2-perturbed lattices. Our method does not formally require assumptions on the correlation measure or the variance behaviour and it retrieves known sharp bounds for neutral integrable systems such as Poisson processes, and also applies to hyperfluctuating systems.