The broad scientific and technological importance of crystallization has led to significant research probing and rationalizing crystal nucleation processes. Previous work has generally neglected the possibility of the molecular-level dynamics of individual crystal nuclei coupling to local structures. However, recent experimental work has conjectured that this can occur. Therefore, to address a deficiency in scientific understanding of crystallization, we have probed the nucleation of prototypical single and multicomponent crystals (specifically, ice and mixed gas hydrates). We establish that local structures can bias the evolution of nascent crystal phases on a nanosecond time scale by, for example, promoting the appearance or disappearance of specific crystal motifs and thus reveal a new facet of crystallization behavior. Moreover, we demonstrate structural biases are likely present during crystallization processes beyond ice and gas hydrate formation. Structurally biased dynamics are a lens for understanding existing computational and experimental results while pointing to future opportunities.