Neocortex-wide neural activity is organized into distinct networks of areas engaged in different cognitive processes. To elucidate the underlying mechanism of flexible network reconfiguration, we developed connectivity-constrained macaque and human whole-cortex models. In our model, within-area connectivity consists of a mixture of symmetric, asymmetric, and random motifs that give rise to stable (attractor) or transient (sequential) heterogeneous dynamics. Assuming sparse low-rank plus random inter-areal connectivity constrained by cognitive networks' activation maps, we show that our model captures key aspects of the cognitive networks' dynamics and interactions observed experimentally. In particular, the anti-correlation between the default mode network and the dorsal attention network. Communication between networks is shaped by the alignment of long-range communication subspaces with local connectivity motifs and is switchable in a bottom-up salience-dependent routing mechanism. Furthermore, the frontoparietal multiple-demand network displays a coexistence of stable and dynamic coding, suitable for top-down cognitive control. Our work provides a theoretical framework for understanding the dynamic routing in the cortical networks during cognition.