Gravity is described by General Relativity, which is a classical theory. But at very small scales, nature is described by quantum theory, not by the continuity assumed in GR. So far, we have failed to reconcile these two theories. To date, we have no proof of the descreteness of space time. In an ideal world (or universe), GR should describe gravity at any scale, but unfortunately it fails at small scales, giving infinity as the result to many problems.
In addition, you may think more about what we mean by an object at these small scales. If you think of the problem in terms of a grain of sand "jumping" from one side of a spacetime box to another, you are (incorrectly) thinking classically, rather than thinking in quantum mechanical terms.
An associated area that might help you is to read about quantum tunnelling in which the object is treated in wave terms at small scale regimes.