We study the Drell-Yan cross section differential with respect to the transverse momentum of the produced lepton pair. We consider data with
moderate invariant mass $Q$ of the lepton pair, between 4.5 GeV and 13.5 GeV, and similar (although slightly smaller) values of the transverse momentum $q_T$. We approach the problem by deriving predictions based on standard collinear factorization, which are expected to
be valid toward the high-$q_T$ end of the spectrum and to which any description of the spectrum at lower $q_T$
based on transverse-momentum dependent parton distributions ultimately needs to be matched. We find that the collinear framework
predicts cross sections that in most cases are significantly below available data at high $q_T$. We discuss additional perturbative and
possible non-perturbative effects that increase the predicted cross section, but not by a sufficient amount.