I am not sure whether this is what you are looking for, but here's a paper discussing Einstein’s paper “Explanation of the Perihelion Motion of Mercury from General Relativity Theory”:

http://www.wbabin.net/eeuro/vankov.pdf
You will recognize the simplest post-Newtonian terms (for this special case).

I am not sure if this is what you are looking for; I guess instead of "slow motion + strong gravitational field" you are interested in "fast motion + weak gravitational field"; anyway - the starting point is always the geodesic e.o.m.