  We give a characterization for asymptotic dimension growth. We apply it to
CAT(0) cube complexes of finite dimension, giving an alternative proof of N.
Wright's result on their finite asymptotic dimension. We also apply our new
characterization to geodesic coarse median spaces of finite rank and establish
that they have subexponential asymptotic dimension growth. This strengthens a
recent result of J. Spakula and N. Wright.
