  We address spin transport in the easy-axis Heisenberg spin chain subject to
integrability-breaking perturbations. We find that spin transport is
subdiffusive with dynamical exponent $z=4$ up to a timescale that is
parametrically long in the anisotropy. In the limit of infinite anisotropy,
transport is subdiffusive at all times; for large finite anisotropy, one
eventually recovers diffusion at late times, but with a diffusion constant
independent of the strength of the integrability breaking perturbation. We
provide numerical evidence for these findings, and explain them by adapting the
generalized hydrodynamics framework to nearly integrable dynamics. Our results
show that the diffusion constant of near-integrable interacting spin chains is
generically not perturbative in the integrability breaking strength.
