  We perturbatively calculate classical radiation in Yang-Mills theory and
dilaton gravity, to next-to-leading order in couplings. The radiation is
sourced by the scattering of two relativistic massive scalar sources with the
dynamical effect taken into account, corresponding to the post-Minkowskian
regime in gravity. We show how to arrange the Yang-Mills radiation such that
the duality between colors and kinematics is manifest, including the three-term
Jacobi identity. The search for duality-satisfying expressions exploits an
auxiliary bi-adjoint scalar theory as a guide for locality. The double copy is
obtained by replacing the color factors in Yang-Mills with kinematic
counterparts, following Bern-Carrasco-Johansson construction in S- matrix. On
the gravity side, the radiation is directly computed at the third
post-Minkowskian order with massive sources. We find perfect agreement between
observables in dilaton gravity and the Yang-Mills double copy. This
non-trivially generalizes the leading-order rules by Goldberger and Ridgway.
For the first time, the kinematic Jacobi identity appears beyond field-theory
S-matrix, suggesting that the color-kinematics duality holds more generally.
Our results offer a path for simplifying analytical calculations in
post-Minkowskian regime.
