The hydrodynamical scalings of many discrete-velocity kinetic models lead to a small-relaxation time behavior governed by the corresponding Euler type hyperbolic equations or Navier-Stokes type parabolic equations. Using as a prototype a simple discrete-velocity model of the Boltzmann equation we develop a class of central schemes with the correct asymptotic limit that work with uniform second order accuracy with respect to the scaling parameter. Numerical results for both the fluid-dynamic limit and the diffusive limit show the robustness of the present approach.