  We compute the effect of scattering gravitational radiation off the static
background curvature, up to second order in Newton constant, known in
literature as tail and tail-of-tail processes, for generic electric and
magnetic multipoles. Starting from the multipole expansion of composite compact
objects, and as expected due to the known electric quadrupole case, both long-
and short-distance (UV) divergences are encountered. The former disappears from
properly defined observables, the latter are renormalized and their associated
logarithms give rise to a classical renormalization group flow. UV divergences
alert for incompleteness of the multipolar description of the composite source,
and are expected not to be present in a UV-complete theory, as explicitly
derived in literature for the case of conservative dynamics. Logarithmic terms
from tail-of-tail processes associated to generic magnetic multipoles are
computed in this work for the first time.
