The asymptotic behavior of solutions to a reaction-diffusion system with nonlinear boundary conditions is discussed. It is assumed that the boundary values are controlled by a positive parameter $\epsilon $ and that the boundary conditions reduce to the homogeneous Neumann boundary ones if $\epsilon $ tends to 0. Under appropriate conditions it is shown that for each small $\epsilon $ there exists an inertial manifold $\mathcal{M}_\epsilon $, that is, a finite-dimensional Lipschitz (or $C^1 $) manifold which is invariant and attracts every solution exponentially. Moreover, it is proved that as $\epsilon \to 0$ the manifold $\mathcal{M}_\epsilon $ converges to $\mathcal{M}_0 $, which is the one for the homogeneous Neumann boundary conditions. The dynamics on the manifold are investigated through the reduced ordinary differential equation on it, called the inertial form, for specific cases; for instance, a specific example shows that the boundary values induce a relaxation-oscillating periodic motion in the manifold while every solution converges to a steady state in the case $\epsilon = 0$.