I'll tell you how I picture them (and I warn you, there are problems with this picture, but it is the best I can do interpretating rather complicated mathematics)
Gravitons, by themselves are no different than any other gauge particle. Taken by themselves, if you could put them on mass shell, they'd propogate just like any other free particle in whatever ambient spacetime with whatever curvature you want with some appropriate equation of motions/geodesics for a massless, spin2 particle.
Now they won't ordinarily arise in that sort of free field case, usually the situation we have is two point masses that interact through the exchange of a virtual graviton. Here again, by itself the graviton is again no different than any other gauge particle. What is however different, is that the *interaction* between the graviton and the point particles, as well as the graviton on graviton interaction (arising from higher order diagrams) will (if you could sum up all the diagrams) induce a backreaction on the actual metric tensor itself, causing a curvature change. Thats exactly what you want. A coherent state of gravitons can and will create dynamics for the gravitational field or alternatively in the more familiar GR language, create the dynamics for how Geometry changes and evolves.
In fact, to answer your original question, the only case where we actually know how to solve anything, is exactly the flat space case. Here you take the original ambient space to also be flat, and we require the point masses to be completely tiny (or far away) such that the interaction is completely negligable, leaving the flat space intact to good approximation. Or in other words, the interaction to preserve zero curvature must be trivial, just like in normal GR.
