  The parton quasi-distribution functions (QDFs) of Ji have been found by
Radyushkin to be directly related to the transverse momentum distributions
(TMDs), to the pseudo-distributions, and to the Ioffe-time distributions
(ITDs). This makes the QDF results at finite longitudinal momentum of the
hadron interesting in their own right. Moreover, the QDF-TMD relation provides
a gateway to the pertinent QCD evolution, with respect to the resolution scale
Q, for the QDFs. Using the Kwieci\'nski evolution equations and well
established parameterizations at a low initial scale, we analyze the QCD
evolution of quark and gluon QDF components of the proton and the pion. We
discuss the resulting breaking of the longitudinal-transverse factorization and
show that it has little impact on QDFs at the relatively low scales presently
accessible on the lattice, but the effect is visible in reduced ITDs at
sufficiently large values of the Ioffe time. Sum rules involving derivatives of
ITDs and moments of the parton distribution functions (PDFs) are applied to the
ETMC lattice data. This allows us for a lattice determination of the
transverse-momentum width of the TMDs from QDF studies.
