Sr₂RuO₄ and AFe₂As₂ (A = K, Rb, Cs) are strongly correlated Hund’s metals that lie close to a strain-driven electronic instability, primarily associated with the dxy-orbital character of their band structure. Both systems exhibit the characteristic properties of multiband superconductors. The superconducting transition is accompanied by a distortion of the crystal lattice, which determines the stress dependence of the transition temperature T_c and provides insight into the underlying superconducting state. We have studied the lattice distortion of Sr₂RuO₄ and AFe₂As₂ single crystals by measuring their thermal expansion between 20 mK and 4 K. In contrast to measurements under finite uniaxial pressure, thermal expansion does not break the crystal symmetry but allows the stress-dependent entropy to be tracked in the superconducting state. The AFe₂As₂ crystals exhibit the commonly observed behavior in which the anisotropy of the lattice distortion remains approximately constant below . In Sr₂RuO₄, however, the response changes from nearly isotropic close to Tc to strongly anisotropic upon cooling. This unusual evolution of the stress response suggests a pronounced temperature-dependent redistribution of the superconducting response among different bands in Sr₂RuO₄, consistent with the presence of strongly band-selective superconductivity in a system close to a strain-tuned electronic instability.