How do these gravitons move around in space-time?
What happens if you took two gravitons and you collide them with each other?
It appears that your explanations and establishment of problem are developed within the tradition of Newtonian mechanics conception. In the framework of Newtonian mechanics problem can be solved analytically only for local interactions. Mathematical formulas of Newton Law (two bodies with masses interact), or Coulomb Law (two bodies having rest charges interact), or Ampere Law (two conductors having moving charges interact) are rather determinate. Therefore, such mathematical formulas have no ground in the reality. Indeed, if the third body with mass appears in the closed system, one gets three-body-problem.
There is an assumption within the modern theoretical physics that gravitation and antigravitation are not from our universe. Graviton "serves as a messenger particle for gravitation" is from outside of our world.
Following from what has been said I think it should be better to represent "very-very simplified at a level of profanation" the point of view that universe constitutes of rational and irrational parts (or there is this realm and there is the other realm).
Rene Descartes proposed philosophically based views on the duality of universe more than 400 year ago. He assumed that this realm is "extension in length, breadth, and thickness constitutes the nature of corporeal substance". The other realm is interaction of ideal vortices.
Then Newton who believed in Pantocrator did not believe in a duality of universe. "Hypotheses non fingo": Newton was mocking at Descartes hypothesis that ideal vortices in the ideal (other) realm of universe are inherent of every object in this realm of universe. Ideal vortices interact with each others in the vorticity-realm and create visible and invisible phenomena in the extension-realm. The vortical motion of Cartesian particles embarrassed Newton; Newton separate particles from vortices. Newton kept the particles and denied the vortices.
But Descartes idea that there are ideal vortices has been upraised in modern theoretical physics.
Compare: string is the object which has no any physical properties (mass, charge, etc.) except of length [meter]. "Linear" string is the mathematical basis of this "extension realm" in the duality universe model by Descartes.
Closed string has no any physical characteristics (mass, charge,...) and cannot be represented with length [meter]. Closed string is the mathematical basis of the "vorticity realm" in the duality universe model by Descartes.
Herewith a simplified dual model of universe has eight dimensions: x, y, z dimensions by Descartes; phi, lambda tetta dimensions by Euler; t dimension by Einstein; u0=dtau/dt dimension by Cartan-Vlasov (see Section 7. Cauchy's problem in Vlasov A.A. Many-particle theory. Moscow-Leningrad, State publishing technical and theoretical literature (GITTL), 1950. 348pp. (in Rus.) or Vlasov A.A. Many-particle theory and its application to plasma, Trans, Russian Gordon and Breach Science Publishers. Inc., New York, 1961.). Further reading Chapter 11. Covariant statistical equations and temperature distribution in eight-dimensional space of line elements (Finsler space) in Vlasov A.A. Nonlocal statistical mechanics. Moscow: Nauka, 1978. 264pp. (in Russian).