The EFE - The 'Mother' of all equations of motion in the universe? This question addresses the scope of general relativity.
Scientists and engineers solve all sorts of physical problems in all sorts of fields and often to solve these problems a 'system' is defined including inputs, outputs, inter-relationships of variables and constraints, and from this definition the 'equations of motion' are written - specific to that system.
So my question is can the Einstein Field Equations (EFE) of General Relativity lead to the same equations of motion for these specific systems in all cases (discounting for the moment quantum mechanical systems)? In other words: 
Are the EFE of General Relativity the 'Mother' of all equations of motion in the universe - outside of quantum mechanical systems? - And therefore a more generalized way of writing equations of motion for all systems?
If this is true, then from the field equations, can one derive Maxwell's equations?
 A: No.
The Einstein field equations are the equation of motion for the metric (i.e. gravity) in the Einstein-Hilbert action.
If you add other dynamical fields to the action, you not only change the stress-energy tensor appearing in the EFE, but you also have to vary the action with respect to the new fields to obtain e.o.m. for them.
A: I don't agree with the previous answer.
Firstly, the OP's question isn't about the lagrangian formulation, it's about the Einstein equation :
\begin{equation}\tag{1}
G_{\mu \nu} + \Lambda \, g_{\mu \nu} = -\; \kappa \, T_{\mu \nu}.
\end{equation}
Secondly, there are stress-tensors that can't be derived from an action : fluids tensors (especially with viscosity), for example.
Once you have added all contributions to energy-momentum to the right part of equ. (1), then all the local geometrical properties of spacetime are determined.  And since gravity weights everything, that equation contains local conservation of all forms of energy-momentum.  From Bianchi identity, you can derive the local conservation of energy-momentum, which is built-in the Einstein equation :
\begin{equation}
\nabla_{\mu} \, T^{\mu \nu} = 0.
\end{equation}
From that, you can deduce Maxwell's equation, or the Klein-Gordon equation, or the Navier-Stokes equation (the only case I know which may be problematic is the Dirac equation, because of some algebraic subtleties).
So in a sense, yes the Einstein equation is the "Mother of all equations".
The only things that the Einstein equation isn't able to predict are the global properties of spacetime : number of dimensions, topologic features (spatially open or closed universe, boundary conditions, etc).
There's a very good discussion about all these in the huge MTW book (Misner, Thorne & Wheeler).
EDIT : I'm adding a nice citation from MTW's book (page 475), which says alot :

The Maxwell field equations are so constructed that they automatically fulfill and demand the conservation of charge ; but not everything has charge.  The Einstein field equation is so constructed that it automatically fulfills and demands the conservation of momentum-energy ; and everything does have energy.  The Maxwell field equations are indifferent to the interposition of an "extrenal" force, because that force in no way threatens the principle of conservation of charge.  The Einstein field equation cares about every force, because every force is a medium for the exchange of energy.
Electromagnetism has the motto, "I count all the electric charge that's here."  All that bears no charge escape its gaze.
"I weigh all that's here" is the motto of spacetime curvature.  No physical entity escapes this surveillance.

This is why, in a sense, Einstein equation is the Mother of all equations : everything is implied in it, even if it could be difficult to explicitely extract the equations from the "Mother".
I think that this MTW citation is a beautifull resume of relativistic gravitation.  The pure beauty of the "Mother" field equation is all there !
