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The GR field equations, if I remember correctly from the Susskind lectures, include the use of the Lorentz transformations.

So that is the first expression of the equation, and it is equal to the latter part. The latter part also being moving parts?

The Einstein tensor is part of an equality that relates energy and momentum to spacetime curvature. The idea is that accelerated motion in flat space is equivalent to uniform motion in curved space.