In general theory of relativity the Einstein field equations e.g. relate the geometry of space-time with the distribution of one body within it. $$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=\dfrac{8\pi G}{c^4}T_{\mu\nu}.$$
What will be the "Einstein field equations" for two or three bodies?
They are already included, that is a field equation. So the energy-momentum tensor on the right must include the distribution of energy (matter) and momentum in your model, all of it. Normally it is used the other way around, one takes a particular form of the energy momentum tensor to model a single "point"-like particle, by for example making the energy density into a delta function evaluated at the particles wordline.
$T_{\mu\nu}$ represents the sources. So, if you have n bodies, you must take them into account into $T_{\mu\nu}$.
The approach is totally analogue to the definition of the sources for the Maxwell's equations: also in that case, there is no formal difference between a pointlike charge and a distribution of charge density. You just have to properly write the source term.
The Einstein-Infeld-Hoffman equations, derived in 1938, describe the motion of $N$ gravitating point masses in General Relativity. They are an expansion in powers of $1/c$, so they are useful when the particles are not highly relativistic. The leading term (with no power of $1/c$) is simply Newtonian gravity.
