  We study the finite-momentum spin dynamics in the one-dimensional XXZ spin
chain within the Ising-type regime at high temperatures using density
autocorrelations within linear response theory and real-time propagation of
nonequilibrium densities. While for the nonintegrable model results are well
consistent with normal diffusion, the finite-size integrable model unveils the
coexistence of anomalous and normal diffusion in different regimes of time. In
particular, numerical results show a Gaussian relaxation at smallest nonzero
momenta which we relate to nonzero stiffness in a grand canonical ensemble. For
larger but still small momenta normal-like diffusion is recovered. Similar
results for the model of impenetrable particles also help to resolve rather
conflicting conclusions on transport in integrable Mott insulators.
