  In contrast to our understanding of density field tracers, the modelling of
direct statistics pertaining to the cosmic velocity field remains open to
significant opportunities for improvement. The lack of accurate modelling for
the non-linear domain of pairwise velocities restricts our capacity to fully
exploit the information encoded in this observable. We present a robust
approach for modelling the mean infall velocities, $v_{12}(r,a)$, with broad
applicability spanning sub-megaparsec scales and cosmologies extending beyond
the standard LCDM paradigm. Our approach involves solving the full
pair-conservation equation using accurate non-linear power spectrum
descriptions. To assess the robustness of our model, we extend it to
cosmologies beyond the standard LCDM, in particular, the Hu-Sawicki
$f(R)$-gravity and Dvali-Gabadadze-Porrati (DGP) modified gravity models.
Remarkably, our predictions for pairwise velocities of dark matter particles at
kilo-parsec scales exhibit excellent agreement with N-body simulations
throughout the entire dynamical range ($0.1 \lesssim \xi \lesssim 1000$, or
$r\geq0.4$Mpc/h). Furthermore, we show that different gravity models leave
distinct signatures in the shape and dynamics of the mean pairwise velocities,
providing a potent test of cosmological gravity laws.
