As long as you appreciate these are all deeply flawed "lies to children" mental pictures, and need not "represent" a consistent "story"...
Nothing of these things happen, or even remotely so, any more than a snake biting its tail represents cyclic organic compounds (another celebrated dream metaphor).
The brick thrown from boat to boat metaphor asks you to suspend your disbelief when the brick traveling from boat to boat sucks momentum from the recipient (absorber) boat and pulls it towards the emitter boat. This does not happen in conventional mechanics that you have build intuition for, but in quantum mechanics, a deeply counterintuitive theory, where you mostly follow the math.
Such pictures, including terms like photons or gravitons, are cartoon mnemonics for well-understood and applied calculational recipes: they are shorthand for mathematical expressions in perturbation theory, a calculational technique that yields good answers. Highly perceptive people have often been led astray by them.
In electromagnetism, two magnets emit and absorb gazillions of low energy virtual photons at and from each other, resulting in a mutual attraction (if they are aligned properly).
In quantum gravity, a speculative/hypothetical type of theory, an apple and the earth emit and absorb gagillions of "soft" gravitons at and from each other, creating a mutual attraction, classically understood as gravitational field attraction. Repulsion is not possible here.
In electromagnetism, (quantum) atomic transitions are described by photons than by emissions and absorption of classical electromagnetic waves.
In gravity, microscopic quantum phenomena are not observable directly (or have been observed), so you observe gravitational classical fields from and to everything that has energy (including light, and photons) or mass; or classical gravitational waves from fast-moving massive black hole or neutron star binaries, etc.
Because (in some highly technical way outranging our scope here) gravity is "weak", in comparison, it is freakishly hard/impossible to observe the type of quantum microscopic transition you are envisioning. So your pre-fab paragraph needs heavy deconstructive revision,
- When a mass or clump of energy is in any state, it emits an indefinite number of real or virtual gravitons, packets of energy and momentum which thus carry that Gravity through space at the speed of light until it encounters another clump, spaceship, nebula, or photon which absorbs it, picking up the energy and momentum transferred.
however, I'm not quite sure what you were going to do with that picture. Unlike real photons which are routinely observable in the lab by several gadgets and techniques, single gravitons are not even remotely observable; F Dyson is famous for having doggedly sought a proof for this.
When theorists yap breathlessly about gravitons, they involve them in elaborate quantum calculations which might contribute to indirect shifts of recondite quantities possibly providing "smoking guns" for their (the gravitons') existence.
But it is quite hard to develop intuition on something unobserved and arguably disconnected from observable phenomena.