Context. The stellar mass-star formation rate (M⋆ ‑ SFR) plane is an essential diagnostic to separate galaxy populations. However, we still lack a clear picture of how galaxies move within this plane along cosmic time. Aims. This study aims to provide an observational description of galaxy migrations in the M⋆ ‑ SFR plane based on the reconstructed star formation histories (SFH) of a sample of galaxies at redshift z < 4. Ultimately, this study seeks to provide insight into physical processes driving star formation. Methods. We used data from the COSMOS field, which provides extensive multi-wavelength coverage. We selected a sample of 299131 galaxies at z < 4 with the COSMOS-Web NIRCam data at a magnitude of mF444W < 27 over a large area of 0.54 deg2. We utilized the SED modeling code CIGALE, which incorporates non-parametric SFHs, to derive the physical properties and reconstruct the SFHs of this galaxy sample. To characterize the SFHs and interpret the galaxies' movements on the M⋆ ‑ SFR plane, for each galaxy we also defined a migration vector in order to track the direction (ΦΔt[deg]) and velocity norm (rΔt[dex/Gyr]) of the evolutionary path over the M⋆ ‑ SFR plane. We quantified the quality at which these migration vectors can be reconstructed using the HORIZON-AGN cosmological hydrodynamical simulation. Results. We find that galaxies within the main sequence exhibit the lowest amplitude in their migration and a large dispersion in the direction of their movements. We interpret this result as galaxies oscillating within the galaxy main sequence. By using their migration vectors to find the position of main-sequence progenitors, we obtained that most of the progenitors were already on the main sequence as defined one billion years earlier. We find that galaxies within the starburst or passive region of the M⋆ ‑ SFR plane have very homogeneous properties in terms of recent SFH (< 1 Gyr). Starburst galaxies assembled half of their stellar mass within the last 350 Myr, and this population originates from the main sequence. Galaxies in the passive region of the plane show a homogeneous declining SFH over the full considered redshift range. We identified massive galaxies already in the passive region at 3.5 < z < 4, and their number density increases continuously with cosmic time. The progenitors of passive galaxies are distributed over a large range of SFRs, with less than 20% of passive galaxies being starburst 1 Gyr earlier, thus shedding light on rapid quenching channels. Conclusions. Using reconstructed SFHs up to z < 4, we propose a coherent picture of how galaxies migrate over cosmic time in the M⋆ ‑ SFR plane, highlighting the connection between major phases in the SFH.