  We study the ballistic transport in integrable lattice models, i.e., the spin
XXZ and Hubbard chains, close to the noninteracting limit. The stiffnesses of
spin and charge currents reveal, at high temperatures, a discontinuous
reduction (jump) when the interaction is introduced. We show that the jumps are
related to the large degeneracy of the parent noninteracting models. These
degeneracies are properly captured by the degenerate perturbation calculations
which may be performed for large systems. We find that the discontinuities and
the quasilocality of the conserved current in this limit can be traced back to
the nonlocal character of an effective interaction. From the latter observation
we identify a class of observables which show discontinuities in both models.
We also argue that the known local conserved quantities are insufficient to
explain the stiffnesses in the Hubbard chain in the regime of weak interaction.
