Embryogenesis--generation of functional forms--entails coordinated cell motion (morphogenesis), intercellular communications via morphogen patterns, and cell fate decisions. Morphogenesis and patterning have traditionally been studied separately, and how cell movement affects cell fates remains unclear. Traditional models of pattern formation deal mostly with static tissues, preventing the rationalization of increasingly available spatiotemporal data of morphogens and flows in remodeling tissues. We present a theoretical framework for pattern formation in motile cell environments by describing the dynamics of morphogen exposure felt by moving cells (Lagrangian frame) rather than at fixed laboratory coordinates (traditional Eulerian frame). This cell frame description reveals how morphogenetic motifs such as multicellular attractors and repellers (i.e., the Dynamic Morphoskeleton) and convergent extension flows act as barriers and enhancers to diffusive morphogen transport, revealing a robust synergy between morphogenesis and intercellular signaling. We apply our framework to standard models for dynamic cell fate bifurcations and induction and to experimental data from avian gastrulation flows.