The authors investigate the one-dimensional Blume-Emery-Griffiths model with Kawasaki dynamics, using domain-wall arguments (DWA) and Monte Carlo simulations (MCS) (conventional and Gillespie algorithm), after a quench from a disordered state to low temperatures. They observe that this model exhibits domain scaling behaviour, controlled by a universal exponent (x=1/3) as in other dimensions for model B. However, they find also that the critical exponent z is not universal and depends on the coupling constants of the Hamiltonian. The results of DWA are consistent with those of MCS.