The scaling relations between the gas content and star formation rate of galaxies provide useful insights into the processes governing their formation and evolution. We investigated the emergence and the physical drivers of the global Kennicutt-Schmidt (KS) relation at 0.25 ≤ z ≤ 4 in the cosmological hydrodynamic simulation NEWHORIZON, capturing the evolution of a few hundred galaxies with a resolution down to 34 pc. The details of this relation vary strongly with the stellar mass of galaxies and the redshift. A power-law relation Σ_SFR ∝ Σ_gas^a with a ≈ 1.4, like that found empirically, emerges at z ≈ 2 − 3 for the more massive half of the galaxy population. However, no such convergence is found in the lower-mass galaxies, for which the relation gets shallower with decreasing redshift. At galactic scales, the star formation activity correlates with the level of turbulence of the interstellar medium, quantified by the Mach number, rather than with the gas fraction (neutral or molecular), confirming the conclusions found in previous works. With decreasing redshift, the number of outliers with short depletion times diminishes, reducing the scatter of the KS relation, while the overall population of galaxies shifts toward low densities. Our results, from parsec-scale star formation models calibrated with local Universe physics, demonstrate that the cosmological evolution of the environmental (e.g., mergers) and internal conditions (e.g., gas fractions) conspire to shape the KS relation. This is an illustration of how the interplay of global and local processes leaves a detectable imprint on galactic-scale observables and scaling relations.