  Lattice QCD offers the possibility of computing parton distributions from
first principles, although not in the usual $\overline{MS}$ factorization
scheme. We study in this paper the evolution of non-singlet parton distribution
functions (PDFs) in the short-distance factorization scheme which notably
arises in lattice calculations in the pseudo-distribution approach. We provide
an assessment of non-perturbative evolution of PDFs from already published
lattice matrix elements, and show how this evolution can be used to reduce the
fluctuation of the lattice data. We compare our result with expectations
obtained thanks to a perturbative matching to $\overline{MS}$. By highlighting
the limitations of the current computations, we advocate for a new strategy
using lattice calculations in small volume.
