I'm not sure what you're asking, exactly. Please ask again, if this doesn't help:

Photons have "gravitational charge", in that they both feel other bodies' gravitational influence, and induce a gravitational influence on other bodies. One sees the "feeling" part in observations of gravitational lensing even around our own sun; the generating part is harder to demonstrate, but consider https://en.wikipedia.org/wiki/Kugelblitz_(astrophysics)

In General Relativity, we explain the gravitational charge in terms of spacetime curvature; light both generates curvature (because it has momentum) and responds to curvature (because of the universality of free fall).

An isolated photon (or even a non-extreme classical beamlike pulse of light) generates so little curvature that we can safely ignore it in an otherwise Minkowski (flat) spacetime; however, in both cases if it traverses a region of spacetime that is not flat, the isolated photon and classical pulse will respond accordingly.

More technically, light contributes to the matter tensor, whose components encode the flux of momentum from one direction to another. Although light has no rest mass, it does have energy-momentum (in special relativity, E = pc; additionally, the Planck-Einstein relation for photon energy E = h\nu, where h is Planck's constant and \nu is the frequency of the photon). So a higher frequency photon contributes more to the stress-energy tensor than a lower frequency photon, and even the lowest-frequency imaginable contributes more than vacuum. (Aside: General Relativity runs into localization problems with extremely low frequency photons if one piles up a lot of them in a region of space comparable in diameter to their wavelengths; they should mutually gravitate, and possibly collapse, but exactly how is so hard to show that probably nobody's done so successfully. How do gravitational interactions among the photons change their wavelengths? (This is an aspect of quantum gravity; there are related practical problems with Hawking radiation, especially near final evaporation, for instance)).

> particles would be on their own timeline orthogonal to all others

You can contrive a "dimpled" mostly-flat spacetime in which small masses are so widely separated that the curvature they induce is not felt by their neighbours or vice-versa. The contrivance is not especially awful, because we do something similar with "swiss cheese" models of cosmology, where the holes in the cheese are galaxy clusters generating a non-expanding or even collapsing metric (like Schwarzschild or Lemaître-Tolman-Bondi) and the body of the cheese is an expanding Robertson-Walker metric. Already separated holes "float" in the cheese, and the expansion just separates them with ever more cheese, without interfering in the evolution of the galaxies etc. in the holes. Holes which are too close together coalesce into a single larger hole. The spacetime curvature of an expanding "swiss cheese" is extremely strong, but we can turn off the expansion and it is similar to the "dimpled" mostly-flat one; in fact you could model the "dimples" as tiny patches of Schwarzschild space stitched into Minkowski space [1]. The "holes" that are sufficiently separated by the non-growing cheese would stay separated forever. In all these cases, the worldlines of the centres of masses of the holes are essentially entirely timelike.

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[1] This kind of "stitching" using junction conditions is straightforward if extremely complicated and subject to a number of conditions/assumptions. Misner, Thorne & Wheeler's _Gravitation_ has a chapter on Israel junction conditions, and various authors notably Darmois, O'Brien, Synge, Bonnor and Raju have catalogued stitchings of different types of metrics into one another. Most of these would likely require a trip to sci-hub, as they were written largely in the 1970s to 1990s. Unfortunately, wikipedia seems very sparse on this topic (see for example the Inhomogenous cosmology entry, which is desperate for some love [<- possible note to self]).