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Recent observations of global velocity gradients across and along molecular filaments have been interpreted as signs of gas accreting onto and along these filaments, potentially feeding star-forming cores and proto-clusters. The behavior of velocity gradients in filaments, however, has not been studied in detail, particularly on small scales (< 0.1 pc). In this paper, we present MUFASA, an efficient, robust, and automatic method to fit ammonia lines with multiple velocity components, generalizable to other molecular species. We also present CRISPy, a Python package to identify filament spines in 3D images (e.g., position-position-velocity cubes), along with a complementary technique to sort fitted velocity components into velocity-coherent filaments. In NGC 1333, we find a wealth of velocity gradient structures on a beam-resolved scale of ~0.05 pc. Interestingly, these local velocity gradients are not randomly oriented with respect to filament spines and their perpendicular, i.e., radial, component decreases in magnitude towards the spine for many filaments. Together with remarkably constant velocity gradients on larger scales along many filaments, these results suggest a scenario in which gas falling onto filaments is progressively damped and redirected to flow along these filaments.