Abstract: The Thouless theory of quantum pumps establishes quantized transport per cycle and determines the conditions for that. When the description is shifted to a moving frame, as prompted by recent experiments on ultracold gases, transported and existing charges mix. Their transformation is encoded in Galilean space and time, but underlying it is one of vector bundles that may be described in a number of ways, that may or may not rely on Bloch theory. In one of them, the transformation mixes strong and weak topological indices of a bundle on a 2-torus; in another one, a 3-torus is at center stage. As a further alternative, the covariance is placed in the context of the Landauer-Büttiker formalism. (Joint work with T. Esslinger and F. Santi)