As it is well known by now, the pre-partonic phase in hadron collisions is successfully described by the Color Glass Condensate (CGC) approach. Previous studies, based on experimental data obtained on a wide range of energies at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC), evidenced that observables related to the dynamics of the collision, i.e. the mean transverse momentum ($\langle p_T \rangle$), the slope of the $\langle p_T \rangle$ dependence on the mass of the hadrons and the average transverse flow velocity obtained from the simultaneous fits of the $p_T$ spectra of the particles with the Boltzmann-Gibbs Blast Wave (BGBW) expression, scale rather well as a function of the square root of the ratio of the particle density over unit of rapidity to the overlapping area of the colliding nuclei ($\sqrt{(dN/dy)/S_\perp}$), the relevant scale in the gluon saturation picture. Results of a similar study extended to strange and multi-strange hadrons, for both proton-proton (\textit{pp}) and heavy-ion (\textit{A-A}) collision systems are presented in the present paper. The similarities and differences in the behaviour of strange relative to multi-strange hadrons and non-strange hadrons are discussed.