  We study the post-Newtonian dynamics of black hole binaries in
Einstein-scalar-Gauss-Bonnet gravity theories. To this aim we build static,
spherically symmetric black hole solutions at fourth order in the Gauss-Bonnet
coupling $\alpha$. We then "skeletonize" these solutions by reducing them to
point particles with scalar field-dependent masses, showing that this procedure
amounts to fixing the Wald entropy of the black holes during their slow
inspiral. The cosmological value of the scalar field plays a crucial role in
the dynamics of the binary. We compute the two-body Lagrangian at first
post-Newtonian order and show that no regularization procedure is needed to
obtain the Gauss-Bonnet contributions to the fields, which are finite. We
illustrate the power of our approach by Pad\'e-resumming the so-called
"sensitivities," which measure the coupling of the skeletonized body to the
scalar field, for some specific theories of interest.
