  We study disordered XXZ spin chains in the Ising phase exhibiting droplet
localization, a single cluster localization property we previously proved for
random XXZ spin chains. It holds in an energy interval $I$ near the bottom of
the spectrum, known as the droplet spectrum. We establish dynamical
manifestations of localization in the energy window $I$, including
non-spreading of information, zero-velocity Lieb-Robinson bounds, and general
dynamical clustering. Our results do not rely on knowledge of the dynamical
characteristics of the model outside the droplet spectrum. A byproduct of our
analysis is that for random XXZ spin chains this droplet localization can
happen only inside the droplet spectrum.
