The internal cavity matrix of globins plays a key role in their biological function. Previous studies have already highlighted the plasticity of this inner network, which can fluctuate with the proteins breathing motion, and the importance of a few key residues for the regulation of ligand diffusion within the protein. In this Article, we combine all-atom molecular dynamics and coarse-grain Brownian dynamics to establish a complete mechanical landscape for six different globins chain (myoglobin, neuroglobin, cytoglobin, truncated hemoglobin, and chains alpha and beta of hemoglobin). We show that the rigidity profiles of these proteins can fluctuate along time, and how a limited set of residues present specific mechanical properties that are related to their position at the frontier between internal cavities. Eventually, we postulate the existence of conserved positions within the globin fold, which form a mechanical nucleus located at the center of the cavity network, and whose constituent residues are essential for controlling ligand migration in globins.