The stability of travelling waves which occur when a nematic liquid crystal is subjected to crossed electric and magnetic fields has been studied previously by (Stewart and Faulkner, Cont. Mech. Thermodyn. (1997)) where conditions on a control parameter q for stability to occur have been given. This article is concerned with the behaviour of the stable perturbations as time increases. For each of the three travelling wave solutions we calculate both the essential spectrum and the eigenvalues and use these to determine the long-term monotonic or oscillatory behaviour of the perturbations. The results are also relevant to liquid crystals subjected to a single field.