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We introduce a double-stranded Kronig-Penney model and analyze its transport properties. Asymmetric fluxes between two strands with suddenly alternating localization patterns are found as the energy is varied. The zero-size limit of the internal lines connecting two strands is examined using quantum graph vertices with four edges. We also consider a two-dimensional Kronig-Penney lattice with two types of alternating layer with δ and δ connections, and show the existence of energy bands in which the quantum flux can flow only in selected directions. [ABSTRACT FROM AUTHOR]