Protein evolution requires sequence changes to alter molecular properties without disrupting the dynamic organization necessary for function. How residue-level networks of correlated motions accommodate such mutations remains poorly understood. Here, we investigate the evolution of correlated conformational dynamics in nonstructural protein 1 (NS1) variants of influenza A viruses that emerged between 1918 and 2004. From molecular dynamics (MD) simulations we built residue networks in which edge weights reflect generalized correlations of conformational motion. The network of correlated dynamics is characterized using centrality analysis. Overall network properties are well-conserved across NS1 variants, indicating evolutionary robustness of the dynamical network. Residues ranked highly across multiple centrality measures form a predominantly hydrophobic cluster in the protein core. These residues exhibited low conformational entropy and high sequence conservation, consistent with rigid, coordinated motions. In contrast, residues exhibiting large differences in betweenness and closeness centrality ranks occupied flexible, solvent-exposed regions surrounding the core network. These residues also showed lower evolutionary conservation. These findings reveal a core-periphery structure in which a rigid hydrophobic network enables long-range dynamic connectivity while more adaptable peripheral regions accommodate evolutionary sequence variation. This architecture provides a mechanistic framework for long-range epistatic interactions without compromising the underlying structural framework.