For most 3+1 dimensional theories containing gravitons they do so in exactly the same way classical gravitational waves do in our universe. (Somewhat more technically: most such graviton theories define them in terms of variations of the Ricci curvature tensor R_{\mu\nu}, much like we can describe a photon in terms of the electromagnetic tensor F_{\mu\nu}.)

So when-and-where does classical gravitational radiation appear?

Spherically symmetric static sources do not generate gravitational waves. However, if we raise a bump on such a source thus breaking both spherical a symmetry and staticity in favour of a dynamical bumpy spheroid -- then with a light-crossing time of the spherical source, the nearby external spacetime will have settled back down to a spherically symmetrical state. The near-region will also return to static (the curvature in the near-region stops varying) while at ever-further removes from the source one can find a perturbation in the curvature there-and-then.

This also works (although it takes much longer than about a light-crossing time) for objects which are slowly-rotating, roughly spherical, but not shrouded in a horizon. Such objects' bumps will eventually flatten, and the flattening is typically faster for more-massive bodies. We see this in the rocky bodies throughout the solar system. Gravitational radiation is shed during the flattening process, but at much lower amplitudes than on a bumpy black hole.

How do black holes get bumps? When something falls onto them. In particular, we study the collision of neutron stars and other black holes onto black holes at LIGO, Virgo, and soon other gravitational wave observatories. The more massive the infaller, the bigger the bump, and the larger the amplitude of the gravitational waves. Indeed, in several observations the black holes are of comparable mass, so they raise bumps on each other, and this can be seen in the multipole wave form.

Back to gravitons. A classical "chirp" of light can be seen as a large number of photons in the theories of quantum electrodynamics or of the Standard Model. A "chirp" of gravitational radiation detected at LIGO can be seen as a large number of gravitons in theories such as perturbative quantum gravity.

The spacetime outside but nearby a spherically symmetrical essentially-non-rotating object is boringly quiet, whether that object is a cold rocky body or a black hole. There won't be gravitational waves of non-negligible amplitude there. But if we induce a large perturbation by breaking that symmetry, the spacetime outside but near the object is much less boring, and filled with gravitational waves. That freshly-dynamical spacetime will eventually settle down, depending on the properties of its sources (the matter configuration), and thus eventually we get the nice quiet spacetime outside (but nearby) the body again. However, that's because the local gravitational perturbations have run away from the local area as gravitational waves.

A rocky body has a solid surface on which a solid bump can rest for very long times (but compare something that can melt or sublimate, deposited onto the surface of such a body). A black hole does not have a solid surface at the horizon: an infalling object passes right through the horizon. Some treatments (putting it very roughly) think about such an object very quickly melting and spreading all over the "surface" of the horizon. This is mathematically convenient sometimes, but conceptually misleading. The horizon cannot support anything -- nothing can rest on it. That's why the time it takes to flatten a bump on a black hole is about the light-crossing time of the black hole.

However, the gravitational radiation comes from the dynamical spacetime outside the horizon at the time the perturbation is raised. Once things settle down, there is just a bigger horizon.

Not covered above: extremely fast black hole rotation, such that we don't have sphericity or staticity in the first place. This doesn't really change the picture much: an infaller raises a bump, the bump is dragged around because of rotation, and settles down. The spacetime outside the rotating black hole with the bump is enormously dynamical, and emits gravitational waves, which fly away from the rotating black hole. In short order, even to observers many galaxies away (e.g. at LIGO), the spacetime around the perturbed rotating black hole will have settled down. Again, if there are gravitons, LIGO-like observers see enormous numbers of them all at once.

Also not covered above: black hole evaporation. We've never detected this, and might not be able to for up to trillions of years (before that we'd need there to have been small primordial black holes older than even any of the electrons in the universe). There is no full answer for what we should expect to see when the amplitude of gravitational radiation is likely to be high (during final evaporation). This is about the only time when what fell into the black hole over the course of its existence is likely to be relevant -- it likely would determine the spectrum of the gravitational radiation, but that radiation would originate in the near region of highly-dynamical spacetime just outside the shrinking horizon.

Penultimately: there are some really different graviton-containing theories that might describe aspects of our universe (even if they are defined for universes with many more spatial dimensions than the 3 that sufficiently describe all our physical observations to date). However, most really different theories have such different large-scale behaviours that there is no hope of connecting them with e.g. the central black hole of our galaxy.

Finally: one could build a tortured metaphor using a steel ball and blowtorch: heat one spot on the ball until it's glowing red, then switch off the blowtorch. It will have a definite localized hot spot -- a temperature bump -- that is visible from some angles but not others. Eventually the ball will thermalize: it will be essentially the same (cooling) temperature from every angle outside it, with no single glowing hotspot. The glow is the emission of a large number of electromagnetic waves (and those are large numbers of photons) carrying away the energetic perturbation on the sphere.