  This paper concerns the diffusion-homogenization of transport equations when
both the adimensionalized scale of the heterogeneities $\alpha$ and the
adimensionalized mean-free path $\eps$ converge to 0. When $\alpha=\eps$, it is
well known that the heterogeneous transport solution converges to a homogenized
diffusion solution. We are interested here in the situation where
$0<\eps\ll\alpha\ll1$ and in the respective rates of convergences to the
homogenized limit and to the diffusive limit. Our main result is an
approximation to the transport solution with an error term that is negligible
compared to the maximum of $\alpha$ and $\frac\eps\alpha$. After establishing
the diffusion-homogenization limit to the transport solution, we show that the
corrector is dominated by an error to homogenization when $\alpha^2\ll\eps$ and
by an an error to diffusion when $\eps\ll\alpha^2$.
  Our regime of interest involves singular perturbations in the small parameter
$\eta=\frac\eps\alpha$. Disconnected local equilibria at $\eta=0$ need to be
reconnected to provide a global equilibrium on the cell of periodicity when
$\eta>0$. This reconnection between local and global equilibria is shown to
hold when sufficient {\em no-drift} conditions are satisfied. The Hilbert
expansion methodology followed in this paper builds on corrector theories for
the result developed in \cite{NBAPuVo}.
