We compute tree level QCD scattering amplitudes in the $k_T$-factorization approach, which implies that incoming partons
are off the mass-shell. To do that, we use a generalisation of the Britto-Cachazo-Feng-Witten (BCFW) recursion relation, which works, in principle,
for any number of legs; we then use these results to perform a study of both Single (SPS) and Double Parton Scattering contributions (DPS)
to the inclusive 4-jet production in the $k_T$-factorization framework at leading order and center of mass energies $\sqrt{s}= 7$ TeV and $13$ TeV.
We discuss the importance of double parton scattering for relatively soft cuts on the jet transverse momenta and find out that symmetric cuts do not suit well
the $k_T$-factorization predictions, because of a kinematic effect suppressing the double parton scattering contribution. We thus propose
the systematic use of asymmetric cuts in future experimental analyses and propose new variables to pin down DPS via the analysis of the
corresponding differential distributions.