1). There is plenty of evidence that gravitational fields, like electromagnetic fields, propagate at the speed of light. LIGO observations of gravitational waves are the latest, and perhaps strongest, evidence. That would suggest that gravitational fields are propagated by some sort of photon-like particles: gravitons.
2). There is also plenty of evidence that the gravitational acceleration vector for a mass is directed almost exactly towards the current, instantaneous location of an attracting mass. That can only be explained by some form of instantaneous transmission of information– which isn’t consistent with a time-lag due to particles that travel at the speed of light. (Extrapolating the orbital trajectory of an attracting mass does not resolve this issue).
3). Additionally, the radial flux of gravitons from a distant mass would decrease with the square of the distance, while the gravitational potential only decreases linearly with distance. Yet the potential determines the observed rate of slowing of clocks and the amount of reduction in the speed of light within gravitational fields. At large distances the graviton flux would be very low, but its effects would be quite strong. This argues against particles as the mediators of gravitational fields.
I believe that I have resolved these contradictions, and that in fact gravity is propagated by graviton particles. In order to resolve the issues presented above, those gravitons have to have some quite specific, and unexpected, properties. I have posted two preprints discussing this on my RG profile: an overview and a more detailed discussion.
I would appreciate discussions of the validity of the three numbered paragraphs above, plus any additional challenges or solutions to the issue.