In which direction do gravitons travel? Do they originate from mass or are they collected by masses? This is a kind of silly question, that mostly goes unanswered: are gravitons detatched from the objects they act on, like photons in free space.
Maybe gravity or gravitons are collected by mass (including black holes), as they are collected by the curve in spacetime caused by the mass. The origin of gravitons in such an arrangement seems phenominal in many respects, but could it be that space itself is just composed of evenly distributed graviton sources? these spill gravitons when a mass distorts it. 
 A: Gravitons are the hypothetical, as yet, exchange particles in a future quantum field theory of gravity in a unified mathematical model of all forces.
Of course theorists search for a unification of forces and the graviton is expected to be the exchanged virtual particle in gravitational interactions viewed as quantum field theories with similar properties to the photon,Z0 and gluon, the other exchanged virtual particles in interactions between particles.
Photons do not appear as real, carrying energy away from their source, except in transitions between energy levels of a potential model holding charges stable, as in an atom or molecule, or in acceleration of charged particles.In the expected analogy of behavior with photons bodies falling into a black hole might , through their acceleration in the fall, produce free gravitons, ( if they exist ),that would behave similar to photons, i.e. for example be trapped in a black hole  inside the horizon.
Now the picture in the last paragraph is sketchy because there is no accepted unification in the quantum field theoretical framework ( except string theory seems to be well on the way of getting there). It is analogous to how the other three forces manifest in nature in the microcosm. In the macroscopic framework of General relativity it is the distortion of space around a mass that creates the "force" of gravity in the limit with Newtonian gravity, and gravitons will only be explainable in this framework once the unification of all four forces gets to a solid mathematical model.

but could it be that space itself is just composed of evenly distributed graviton sources?

Once a solid mathematical model of unified theories is attained this will be answered, i.e how gravitons and space distortions are related. 
I expect virtuality will play a great role in this explanation: virtual gravitons will be exchanged between gravitational sources creating the general relativity energy-momentum tensor. They will not be evenly distributed but according to solutions of the equations, dependent on the strength of the source. This would be in analogy with the field lines created by electric  or magnetic sources: they can be thought as exchanges of virtual photons between the sources distributed according to the solutions of the problem, as the fields are variable depending on geometry. The gravitons from gravitational sources building up space will be virtual.
At the moment gravitational sources appear as distortions of space, in GR, where there is no place for "gravitons".
addition after bump by community:
In general relativity there are predictions of gravitational waves,which need changes of masses (of at least a quadrupole distribution of the change in masses)  to be  radiated . These have been detected by LIGO in the merging of two black holes,  making the analogy of gravitational waves as emergent from gravitons the way electromagnetic waves are emergent from photons closer to reality.
The quadrupole nature at the quantum level necessitates spin 2 for the gravitons, and string theories have  representations which can include it, getting close to a quantum theory of gravity.
