  We compute the classical effective action of color charges moving along
worldlines by integrating out the Yang-Mills gauge field to next-to-leading
order in the coupling. An adapted version of the Bern-Carrasco-Johansson (BCJ)
double-copy construction known from quantum scattering amplitudes is then
applied to the Feynman integrands, yielding the prediction for the classical
effective action of point masses in dilaton gravity. We check the validity of
the result by independently constructing the effective action in dilaton
gravity employing field redefinitions and gauge choices that greatly simplify
the perturbative construction. Complete agreement is found at next-to-leading
order. Finally, upon performing the post-Newtonian expansion of our result, we
find agreement with the corresponding action of scalar-tensor theories known
from the literature. Our results represent a proof of concept for the classical
double-copy construction of the gravitational effective action and provides
another application of a BCJ-like double copy beyond scattering amplitudes.
