In the past, fascinating relations have been uncovered between the scattering amplitudes of a large web of theories. For instance, the so-called double copy connects pions, flavored scalars, gluons, and gravitons by taking different combinations of a small set of building blocks. In this talk, I will consider these relations from the perspective of the equations of motion, following recent work by Cheung and Mangan. In particular, I will consider the extension to the effective field theory (EFT) regime. This leads to new and efficient amplitude relations, connecting the flavor of scalars to the kinematics of gluons at higher orders in the EFT expansion. As a main result, we found strong evidence that the lowest order Yang-Mills amplitudes encode all (tree-level) scattering data for a fully-resumed UV theory that is known to arise in the double copy of some string theories.