When deriving the effective action $\Gamma$ in the background field method, one splits the field $\phi = \phi_b + \phi_f$ into a background (or mean field) $\phi_b$ and fluctuations $\phi_f$, then proceed to integrate out $\phi_f$:
$$ e^{-\Gamma[\phi_b]} = \int\mathcal{D}\phi_f e^{-S[\phi_b+\phi_f]-\frac{\delta \Gamma}{\delta \phi_b} \phi_f}. $$
The background field is a functional of the source $J$:
$$ \phi_b = \frac{\delta W}{\delta J} $$
and, in general, it doesn't coincide with the classical field $\phi_{cl}$ (which satisfies the classical EOM). Nonetheless, it is sometimes assumed (see eg Peskin, Schroeder, eqs 11.55 and 11.58) that $\phi_b$ satisfies the classical EOM so that one can get rid of the linear term in $\phi_f$ in the Taylor expansion of $S[\phi_b+\phi_f]$:
$$ S[\phi_b+\phi_f] = S[\phi_b] + \int\frac{\delta S[\phi_b]}{\delta{\phi_f}}\phi_f + \cdots $$
Other references (see, e.g., eq. 2.91 of Buchbinder et al.'s Effective Action in Quantum Gravity) seem not to assume that $\phi_b$ satisfies the classical EOM.
So here are my questions (for what it's worth, I'm particularly interested in the case of gravity):
When do I have to choose $\phi_b=\phi_{cl}$, i.e. when do I have to assume that $\phi_b$ satisfies the classical EOM?
The importance of the effective action $\Gamma$ is two-fold. For one, it is the generator of 1PI correlation functions. But it can also be used to study the dynamical evolution of $\phi_b$ under the influence of quantum fields via the quantum EOM $\frac{\delta \Gamma}{\delta \phi_b}=0$. If $\phi_b$ also satisfies the classical EOM, then $\phi_b$ is overdetermined and, in particular, there is no new solution to the quantum EOM. What is the meaning of the quantum EOM then?
