ABSTRACT Using several variants of the cosmological simba simulations, we investigate the impact of different feedback prescriptions on the cosmic star formation history. Adopting a global-to-local approach, we link signatures seen in global observables, such as the star formation rate density (SFRD) and the galaxy stellar mass function (GSMF), to feedback effects in individual galaxies. We find a consistent picture: stellar feedback mainly suppresses star formation below halo masses of $M_{\rm H} = 10^{12} \rm \, {\rm M}_{\odot }$ and before $z = 2$, whereas AGN feedback quenches the more massive systems after $z = 2$. Among simba’s AGN feedback modes, AGN jets are the dominant quenching mechanism and set the shape of the SFRD and the GSMF at late times. AGN-powered winds only suppress the star formation rate in intermediate-mass galaxies ($M_{\rm \star } = 10^{9.5 - 10} \rm \, {\rm M}_{\odot }$), without affecting the overall stellar mass-assembly significantly. At late times, the AGN X-ray feedback mode mainly quenches residual star formation in massive galaxies. Our analysis reveals that this mode is also necessary to produce the first fully quenched galaxies before $z=2$, where the jets alone are inefficient. These initially highly star-forming galaxies contain relatively large black holes, likely strengthening the X-ray-powered heating and ejection of gas from the dense, central region of galaxies. Such extra heating source quenches the local star formation and produces a more variable accretion rate. More generally, this effect also causes the break down of correlations between the specific star formation rate, the accretion rate and the black hole mass.