  We introduce a covariant Multipole Expansion for the scattering of a massive
particle emitting photons or gravitons in $D$ dimensions. We find that these
amplitudes exhibit very powerful features such as universality, soft
exponentiation, orbit and spin multipoles, etc. Using ${\rm{SO}}(D)$
representation theory we show that the photon and graviton amplitudes are
related via a new double copy procedure for massive spinning states. All these
features are then promoted to properties of the observables arising in the
classical version of such theories. Focusing on radiation, we provide two main
applications: 1) An exponential Soft Theorem relating conservative effects and
gravitational radiation to all orders in $\omega$; whose leading order directly
leads to the $D=4$ Memory Effect. 2) A classical double copy to evaluate
gravitational radiation from QED Bremsstrahlung, matching previous classical
computations and extending them to spin-quadrupole order.
