  The modified evolution equation for parton distributions of Dokshitzer,
Marchesini and Salam is extended to non-singlet Deep Inelastic Scattering
coefficient functions and the physical evolution kernels which govern their
scaling violation. Considering the x->1 limit, it is found that the leading
next-to-eikonal logarithmic contributions to the momentum space physical
kernels at any loop order can be expressed in term of the one loop cusp
anomalous dimension, a result which can presumably be extended to all orders in
(1-x). Similar results hold for fragmentation functions in semi-inclusive
e^{+}e^{-} annihilation. The method does not work for subleading
next-to-eikonal logarithms, but, in the special case of the F_1 and F_T
structure and fragmentation functions, there are hints of the possible
existence of an underlying Gribov-Lipatov like relation.
