  In this paper we study the multipole expansion of the long-wavelength
effective action for radiative sources in ($d$+1) spacetime dimensions. We
present detailed expressions for the multipole moments for the case of scalar-,
electromagnetic-, and (linearized) gravitational-wave emission. For
electromagnetism and gravity, we derive expressions for the odd-parity,
magnetic-type moments as SO($d$) duals of the ones traditionally used in the
literature. The $d$-dimensional case features a novel set of `Weyl-type'
moments, coupling to the spatial part of the Weyl tensor, which are absent in
three dimensions. Agreement is found in the overlap with previous known
results, notably in the $d \to 3$ limit. Due to its reliance on dimensional
regularization, the results presented here play a crucial role for the further
development of the Effective Field Theory approach to gravitational dynamics,
and in particular for the computation of the gravitational-wave flux, starting
at the third post-Newtonian order.
