Could rocks from Earth have reached the Kuiper belt, or Neptune at least? If so, how?
(I have tracked down the reference that I made in my comment on the question.)
Presumably earth rocks once blasted into space (by volcanism or meteorite impact) gain or lose energy primarily by orbital interactions with planets or moons. Since these interactions will occasionally give enough energy to escape the solar system entirely, presumably anything short of that (Neptune's orbit, the Kuiper belt) is also possible. The questions is if this will happen often enough to create a reasonable expectation that an Earth rock could have collided with any given solar system body.
Various attempts to identify the odds have been made.  In one, researchers simulated the path of fragments from the Chixculub impact, and found that many may have made it to the outer Solar System within 10 million years.
This is the paper in Astrobiology