  We prove an enhanced limit theorem for additive functionals of a
multidimensional Volterra process $(y_t)_{t\geq 0}$. As an application, we
establish weak convergence of the solutions of rough differential equations
(RDE) of the form $$
  dx^\varepsilon_t=\frac 1 {\sqrt \varepsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+g(x_t^\varepsilon)\,d\mathbf{B}_t,$$
and identify their limits as solutions of an RDE driven by a Gaussian field
with a drift coming from the L\'evy area correction of the limiting rough
driver. The equation models a passive tracer in a random field.
  In particular if $h$ is random field such that $h(x, \cdot)$ a
semi-martingale with spatial parameter $x$, we show that the solutions of the
equations $$ dx^\varepsilon_t=\frac 1 {\sqrt \epsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})\,dt+h(x_t^\varepsilon, dt),$$
  converge weakly to that of a Kunita type It\^o SDE $dx_t=G(x_t,dt)$ where
$G(x,t)$ is a semi-martingale with spatial parameters. Furthermore the
$N$-point motions converge.
