  Increasing attention has recently focused on non-traditional dark-matter
production mechanisms which result in primordial dark-matter velocity
distributions with highly non-thermal shapes. In this paper, we undertake an
assessment of how the detailed shape of a general dark-matter velocity
distribution impacts structure formation in the non-linear regime. In
particular, we investigate the impact on the halo-mass and subhalo-mass
functions, as well as on astrophysical observables such as satellite and
cluster-number counts. We find that many of the standard expectations no longer
hold in situations in which this velocity distribution takes a highly
non-trivial, even multi-modal shape. For example, we find that the nominal
free-streaming scale alone becomes insufficient to characterize the effect of
free-streaming on structure formation. In addition, we propose a simple
one-line conjecture which can be used to "reconstruct" the primordial
dark-matter velocity distribution directly from the shape of the halo-mass
function. Although our conjecture is completely heuristic, we show that it
successfully reproduces the salient features of the underlying dark-matter
velocity distribution even for non-trivial distributions which are highly
non-thermal and/or multi-modal, such as might occur for non-minimal dark
sectors. Moreover, since our approach relies only on the halo-mass function,
our conjecture provides a method of probing dark-matter properties even for
scenarios in which the dark and visible sectors interact only gravitationally.
