  The presence of global conserved quantities in interacting systems
generically leads to diffusive transport at late times. Here, we show that
systems conserving the dipole moment of an associated global charge, or even
higher moment generalizations thereof, escape this scenario, displaying
subdiffusive decay instead. Modelling the time evolution as cellular automata
for specific cases of dipole- and quadrupole-conservation, we numerically find
distinct anomalous exponents of the late time relaxation. We explain these
findings by analytically constructing a general hydrodynamic model that results
in a series of exponents depending on the number of conserved moments, yielding
an accurate description of the scaling form of charge correlation functions. We
analyze the spatial profile of the correlations and discuss potential
experimentally relevant signatures of higher moment conservation.
