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The stochastic dynamics of a grain embedded within a turbulent fluid subject to strong gravitational fields can be formulated as a random walk on a Riemannian manifold. Such curvature-weighted walks provide a framework to model the intrinsic variability of accretion onto compact objects. By solving the relevant Fokker-Planck equation on a black hole background, we find the counterintuitive result that the escape probability of a grain is actually higher compared to flat space. This is a consequence of the stretching of radial cells near the event horizon: there is a greater spatial volume for the particle to wander through before being captured. By simulating a large number of grain trajectories, initially distributed on concentric shells with a density profile set by the thin-disc structure equations, we also study particle fluxes through the horizon. Shallower spectral indices emerge at low frequencies relative to flat space, primarily due to time dilation, and steeper ones at high frequencies. We find that Schwarzschild-weighted spectra broadly match observations of low-frequency X-ray variability from systems like Cygnus X-1 in their hard state, suggesting that geometric drifts may be important in describing stochastic accretion processes.