Writing down equations of motions is unreasonable for interacting quantum field theories in 4-dimensional spacetimes.

Perhaps you meant writing down Lagrangians rather than equations of motion?

For QED, the Lagrangian (unlike valid equations of motion for the fields) can be written down, and is probably found in every textbook on quantum field theory.

But this can also be done for canonical gravity; see, e.g., equation (28) on p.22 of

By formal variation, one can obtain classical equations of motion, but for the QED Lagrangian, these (given, e.g., in Wikipedia, without any caveat about their meaning) describe the dynamics of a **single** Dirac particle coupled to a **classical** electromagnetic field. To obtain QED one would have to quantize these equations, but this introduces ultraviolet and infrared divergences that show that the operator versions of the classical equations are inconsistent. The necessary renormalization destroys their validity even perturbatively, where QED is very successful.

Similarly, the Lagrangian for classical gravity (or variations of it) produce equations of motion for the **classical** gravitational field, not for its quantum version. Their quantum interpretation (and indeed all of quantum gravity beyond its semiclassical approximation) is fraught with difficulties. See Chapter B8 (''Quantum gravity'') of my Theoretical Physics FAQ.

There is also a Lagrangian of classical gravity coupled to the Dirac and the electromagnetic field. Its variation produces the Einstein–Maxwell–Dirac equations. They describe the dynamics of a **single** Dirac particle coupled to **classical** gravity and a **classical** electromagnetic field. To obtain quantum gravity one would have to quantize these equations. Nobody knows how to do it. The same problems as in QED arise, but the nonrenormalizability (according to power counting) poses additional problems.