  We extend the evolution mapping approach, originally proposed by Sanchez
(2022) to describe non-linear matter density fluctuations, to statistics of the
cosmic velocity field. This framework classifies cosmological parameters into
shape parameters, which determine the shape of the linear matter power
spectrum, $P_L(k, z)$, and evolution parameters, which control its amplitude at
any given redshift. Evolution mapping leverages the fact that density
fluctuations in cosmologies with identical shape parameters but different
evolution parameters exhibit remarkably similar non-linear evolutions when
expressed as a function of the clustering amplitude. We use a suite of N-body
simulations sharing identical shape parameters but spanning a wide range of
evolution parameters. Using an efficient method for estimating the
volume-weighted velocity field based on the Voronoi tesselation of the
simulation particles, we study the non-linear evolution of the power spectra of
the velocity divergence, $P_{\theta\theta}(k)$, and its cross-power spectrum
with the density field, $P_{\delta\theta}(k)$. By analysing snapshots at
redshifts where the linear matter perturbations have the same amplitude, we
demonstrate that evolution mapping accurately applies to $P_{\theta\theta}(k)$
and $P_{\delta\theta}(k)$. Deviations at small scales can be modelled in terms
of differences in the suppression factor, $g(a) = D(a)/a$, akin to those
observed for the density field. Evolution mapping simplifies the description of
the cosmological dependence of non-linear density and velocity statistics,
streamlining the sampling of large cosmological parameter spaces for the
analysis of cosmological observables.
