  We study a class of systems whose dynamics are described by generalized
Langevin equations with state-dependent coefficients. We find that in the
limit, in which all the characteristic time scales vanish at the same rate, the
position variable of the system converges to a homogenized process, described
by an equation containing additional drift terms induced by the noise. The
convergence results are obtained using the main result in
\cite{hottovy2015smoluchowski}, whose version is proven here under a weaker
spectral assumption on the damping matrix. We apply our results to study
thermophoresis of a Brownian particle in a non-equilibrium heat bath.
