  Nearly ten years ago, Kang, Ma, Qiu, and Sterman derived an evolution
equation for a $Q\bar{Q}$ pair fragmenting into a quarkonium. In this study we
explore the consequence of this evolution for the color-evaporation model,
focusing on $J/\psi$ transverse-momentum ($p_t$) distributions in proton-proton
collisions. We show that, as expected, it softens the spectrum obtained by
fixed-order calculations. While next-to-leading-order calculations strongly
overestimate data at large $p_t$, ours, including the (approximate) $Q\bar{Q}$
evolution and next-to-leading-order cross sections computed with Madgraph, are
in good agreement with experiments. Since our study with the color-evaporation
model shows a significant effect of the $Q\bar{Q}$ evolution at large $p_t$, a
determination of scale-dependent long-distance-matrix elements of
non-relativistic QCD could be necessary. To describe data at small and
intermediate $p_t$, we use the $k_t$-factorization approach, and we argue that
quarkonia data could help constrain unintegrated parton densities.
