If $\phi_q\ll \phi_c$ then $\phi_c$ obeys the classical equations of motion independent of whatever the quantum fluctuations $\phi_q$ are doing.
As a toy model, consider single-particle quantum mechanics[1].
The action on the closed time contour is
$$S[X] = \int_{\mathcal{C}}dt \left[ \frac{1}{2}\dot X^2 - V(X)\right].$$
Split $X(t)$ into $X^+(t)$ and $X^-(t)$ which reside on the forward branch and backward branch of $\mathcal C$ respectively.
The classical $X^{\rm cl}$ and quantum $X^{\rm q}$ fields would be
$$\begin{align}
&X^{\rm cl} (t) = \frac {1}{2} \left( X^+(t) + X^-(t) \right)
\qquad &X^{\rm q} (t) = \frac{1}{2}\left( X^+(t) - X^-(t) \right)
\end{align}
$$
Rewriting the action in terms of the classical and quantum fields:
$$\begin{align}
S[X] &= \int_{\mathcal C}dt\left[ \frac{1}{2}\dot X^2 - V(x) \right]\\
&=\int_{-\infty}^\infty dt \left[ \frac{1}{2}(\dot X^+)^2 - \frac{1}{2}(\dot X^-)^2 - V(X^+) + V(X^-) \right]\\
&=\int_{-\infty}^\infty dt \left[ 2 \dot X^{\rm cl} \dot X^{\rm q} - V(X^{\rm cl} + X^{\rm q} ) + V(X^{\rm cl} - X^{\rm q} )\right]\\
\textrm{(integrate by parts)} &=\int_{-\infty}^\infty dt \left[ -2\ddot X^{\rm cl} X^{\rm q} - V(X^{\rm cl} + X^{\rm q} ) + V(X^{\rm cl} - X^{\rm q} )\right].
\end{align} $$
Now suppose the quantum component $X^q$ is small.
Expanding the potentials yields
$$S[X]\xrightarrow{X^{\rm cl} \gg X^{\rm q} }- \int_{-\infty}^\infty dt \left[ 2X^q \left( \ddot X^{\rm cl} + \frac{ d V } { d X } \biggr\lvert_{X^{\rm cl}} \right)+O((X^{\rm q})^3) \right].$$
In this limit, no matter what the quantum field $X^{\rm q}$ is doing, the action is minimized if the classical field $X^{\rm cl}$ obeys the classical equations of motion
$$\ddot X^{\rm cl} = - \frac{ d V } { d X } \biggr\lvert_{X^{\rm cl} }$$
[1]: This is an example given in Chapter 3 of "Field Theory of Non-Equilibrium Systems" by Alex Kamenev
