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We study pinning and unpinning of superfluid vortices in the inner crust of a neutron star using three-dimensional dynamical simulations. Strong pinning occurs for certain lattice orientations of an idealized, body-centered-cubic lattice and occurs generally in an amorphous or impure nuclear lattice. The pinning force per unit length is ∼10¹⁶ dyn cm^–1 for a vortex-nucleus interaction that is repulsive and ∼10¹⁷ dyn cm^–1 for an attractive interaction. The pinning force is strong enough to account for observed spin jumps (glitches). Vortices forced through the lattice move with a slipstick character; for a range of superfluid velocities, the vortex can be in either a cold, pinned state or a hot, unpinned state, with strong excitation of Kelvin waves on the vortex. This two-state nature of vortex motion sets the stage for large-scale vortex movement that creates an observable spin glitch. We argue that the vortex array is likely to become tangled as a result of repeated unpinnings and repinnings. We conjecture that during a glitch, the Kelvin-wave excitation spreads rapidly along the direction of the mean superfluid vorticity and slower in the direction perpendicular to it, akin to an anisotropic deflagration.