@BioPhysicist I am not sure I understand how your comment relates to my question.

Another way to pose my question. Virtual particles, although not "real" do mediate the exchange of momentum, otherwise there would be no point to introduce them to explain the onset of let us say the electromagnetic force, the latter being very real. So, suppose the pair exists for a tiny interval $\tau$, I'd like to calculate the $\delta p$ they speculatively would gain when exposed to a gravity field $g$ (just intended in the classical way, without resorting to any QFT of gravity) during said $\tau$. While not worrying whether this makes any sense at all. Just want to get to a number.

That is ok with me, but I was more interested to know the result of calculations for the case that the speculative assumption "yes, that pair of mass carrying particle are indeed exposed to gravity during their very brief existence" is taken for true. You know, just to see whether the result would be completely non-sense, or whether it may stimulate deeper reflections.

I just came across this book [Fourier Analysis, Self-Adjointness. Michael Reed, Barry Simon ](elsevier.com/books/ii-fourier-analysis-self-adjointness/re‌​ed/…) offering a free preview of page 19, which is indeed introducing the subject basically on the same lines as you sketched. Though running through the details of the various demonstrations appears quite demanding.

I am not sure whether that is a proof or just a sketch of how yourself would proceed to prove such a theorem. In that case, I am afraid I am not proficient enough to complete that sketch on my own. But sure, somewhere there shall exist a rigorous proof precisely identifying said "certain properties".

though it does have very important implications on the physics of particles as well. Feynman continues on with discussing non zero amplitudes for paths outside the light cone, anti-particles justification, etc.