  In using transverse-momentum-dependent (TMD) parton densities and
fragmentation functions, important non-perturbative information is at large
transverse position $b_T$. This concerns both the TMD functions and their
evolution. Fits to high energy data tend to predict too rapid evolution when
extrapolated to low energies where larger values of $b_T$ dominate. I summarize
a new analysis of the issues. It results in a proposal for much weaker $b_T$
dependence at large $b_T$ for the evolution kernel, while preserving the
accuracy of the existing fits. The results are particularly important for using
transverse-spin-dependent functions like the Sivers function.
