  We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard
potentials, or hard spheres collision kernels. In a perturbative regime around
the global Maxwellian equilibrium, the linearized Boltzmann multi-species
operator $\mathbf{L}$ is known to possess an explicit spectral gap
$\lambda_{\mathbf{L}}$, in the global equilibrium weighted $L^2$ space. We
study a new operator $\mathbf{L^\varepsilon}$ obtained by linearizing the
Boltzmann operator for mixtures around local Maxwellian distributions, where
all the species evolve with different small macroscopic velocities of order
$\varepsilon$, $\varepsilon >0$. This is a non-equilibrium state for the
mixture. We establish a quasi-stability property for the Dirichlet form of
$\mathbf{L^\varepsilon}$ in the global equilibrium weighted $L^2$ space. More
precisely, we consider the explicit upper bound that has been proved for the
entropy production functional associated to $\mathbf{L}$ and we show that the
same estimate holds for the entropy production functional associated to
$\mathbf{L^\varepsilon}$, up to a correction of order $\varepsilon$.
