We investigate the dynamics of tilted cellular/dendritic array patterns that form during directional solidification of a binary alloy when a preferred-growth crystal axis is misoriented with respect to the temperature gradient. In situ experimental observations and phase-field simulations in thin samples reveal the existence of a propagative source-sink mechanism of array spacing selection that operates on larger space and time scales than the competitive growth at play during the initial solidification transient. For tilted arrays, tertiary branching at the diverging edge of the sample acts as a source of new cells with a spacing that can be significantly larger than the initial average spacing. A spatial domain of large spacing then invades the sample propagatively. It thus yields a uniform spacing everywhere, selected independently of the initial conditions, except in a small region near the converging edge of the sample, which acts as a sink of cells. We propose a discrete geometrical model that describes the large-scale evolution of the spatial spacing profile based on the local dependence of the cell drift velocity on the spacing. We also derive a nonlinear advection equation that predicts the invasion velocity of the large-spacing domain, and sheds light on the fundamental nature of this process. The models also account for more complex spacing modulations produced by an irregular dynamics at the source, in good quantitative agreement with both phase-field simulations and experiments. This basic knowledge provides a theoretical basis to improve the processing of single crystals or textured polycrystals for advanced materials.