We examine the asymptotic behaviors of solutions to Hamilton-Jacobi equations while varying the underlying domains. We establish a connection between the convergence of these solutions and the regularity of the additive eigenvalues in relation to the domains. To accomplish this, we introduce a framework based on Mather measures that enables us to compute the one-sided derivative of these additive eigenvalues under different scenarios, including first-order, second-order, and contact first-order equations. Additionally, we provide examples of how this framework can be applied to other settings.