We find that heavy quark transport beyond leading logarithm at weak coupling is intrinsically non-Gaussian: the longitudinal momentum transfer distribution has that are crucial for equilibration dynamics. We show this by computing the leading-order momentum transfer kernel for relativistic heavy quarks in weakly coupled non-Abelian plasmas, matching perturbative momentum transfer on the thermal scale to hard-thermal-loop-resummed soft physics. This is the same structure previously found in strongly coupled holographic plasmas, showing that it is not peculiar to weak or strong coupling, conformality, or supersymmetry. We therefore expect that this is a robust feature that physical quark-gluon plasma should also exhibit.