FeSe stands out among iron-based superconductors due to its extended nematic phase without the onset of long-range magnetic order. While strain-dependent electrical resistivity has been extensively explored to probe nematicity, its influence on magnetotransport properties remains less understood. In this work, we present measurements of the magnetoelastoresistivity in FeSe as a function of temperature and applied magnetic field. Using a minimal multiband Boltzmann model for transport we derive analytical expressions that capture the magnetic behavior of the whole set of experimental data both in the paramagnetic and in the nematic phase. These findings indicate that a multiband framework can robustly describe the magnetoelasto-transport properties in FeSe and arguably in other iron-based superconductors.