In the formulation of QFT using formal functional integrals, as mentioned by Igor in his answer, the Schwinger-Dyson equation becomes an infinite-dimensional differential equation for the partition function. As such, since the partition function is the generating functional for the Green functions, all you are doing is simply grouping the Schwinger-Dyson equations for all Green functions together in a single formula. I also share Igor's viewpoint that the Schwinger-Dyson equations written in terms of the partition function is more natural, because it makes the structure of the equation much more evident.

That being said, there are two nice references that have not been mentioned by the answers given so far. One is the book *Path Integral Methods in Quantum Field Theory* by R. J. Rivers (Cambridge University Press, 1987), where the Schwinger-Dyson equations are a central concept and are formulated in many different ways. Another, which I strongly recommend, is the paper by M. Dütsch and K. Fredenhagen *The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory*, Commun. Math. Phys. **243** (2003) 275-314, [arXiv:hep-th/0211242][1]. In this second reference, it is shown that the Schwinger-Dyson equation is actually a particular case of the quantum correction (due to perturbative renormalization) to all identities that follow from the classical equations of motion, called the *Master Ward Identity* (MWI for short). The quantum BV-equation mentioned in Urs's answer is also a particular case of the MWI, which has a nice algebraic interpretation - to wit, one can see the classical limit of the MWI as the simple fact that the (left-hand sides of) all identities that follow from the classical equations of motion are elements of the ideal of off-shell functionals of field configurations (satisfying certain conditions which we do not need to recall here) which is generated by the Euler-Lagrange operator. At the quantum level, not all identities survive due to renormalization - the ensuing violations are known as anomalies. This means that the "classical" ideal of formal power series in $\hbar$ of functionals generated by the Euler-Lagrange operator is no longer an ideal with respect to the quantum product. However, the most general form for the anomalies is dictated precisely by the MWI. In the case of local gauge (or BRS) symmetries, this leads through the quantum BV equation to the Wess-Zumino consistency conditions and so on.

Just a final word of warning: it must be said that all of the above can be made rigorous *only at the formal perturbative level*. In constructive QFT, the Schwinger-Dyson equations are usually *not* used as a tool to construct a model but are rather derived *rigorously* as a consequence *after* the model has been constructed, since only then can they be made sense of. Analysis of QFT models using the Schwinger-Dyson equations is usually deemed as "non-perturbative" for the equations make no explicit mention of perturbation theory, but making sense of this equations outside perturbation theory is a messy business.

  [1]: https://arxiv.org/abs/hep-th/0211242