It has been observed in the literature that measurements of low-mass Drell–Yan (DY) transverse momentum spectra at low center-of-mass energies s are not well described by perturbative QCD calculations in collinear factorization in the region where transverse momenta are comparable with the DY mass. We examine this issue from the standpoint of the Parton Branching (PB) method, combining next-to-leading-order (NLO) calculations of the hard process with the evolution of transverse momentum dependent (TMD) parton distributions. We compare our predictions with experimental measurements at low DY mass, and find very good agreement. In addition we use the low mass DY measurements at low s to determine the width q s of the intrinsic Gauss distribution of the PB-TMDs at low evolution scales. We find values close to what has earlier been used in applications of PB-TMDs to high-energy processes at the Large Hadron Collider (LHC) and HERA. We find that at low DY mass and low s even in the region of p T / m DY ∼ 1 the contribution of multiple soft gluon emissions (included in the PB-TMDs) is essential to describe the measurements, while at larger masses ( m DY ∼ m Z ) and LHC energies the contribution from soft gluons in the region of p T / m DY ∼ 1 is small.