Chiral Floquet phases support quantized chiral flow along their boundaries even when the bulk is fully localized. In this Letter, we establish a complementary phenomenon, which we call Hilbert-space pumping: under a boundary condition twisted by time translation, a part of the boundary Hilbert space, of quantized dimension, is transferred between the opposite boundaries. The twist is the temporal analogue of Laughlin's flux insertion in the quantum Hall effect. For noninteracting fermions, the number of transferred boundary modes is determined by the winding number, as we show analytically for the boundary dynamics and numerically for the anomalous Floquet Anderson insulator. For interacting systems whose boundary dynamics is a quantum cellular automaton, the transferred dimension is set by the Gross-Nesme-Vogts-Werner (GNVW) index: the twisted boundary Floquet operator is the Floquet operator on a tilted time slice, whose Hilbert-space dimension is multiplied by the index for each period of twist. We derive this relation for matrix product unitaries and test it numerically in random unitary circuits. Applied to the matrix product operators of non-invertible symmetries, the same twisted rank returns the quantum dimension of the symmetry operator in addition to its flow index.