We investigate the temporal evolution of a ferromagnetic system of Ising spins evolving under Kawasaki dynamics from a random initial condition, in spatial dimensions one and two. We examine in detail the asymptotic behaviour of the two-time correlation and response functions. The linear response is measured without applying a field, using a recently proposed algorithm. For the chain at vanishingly small temperature, we introduce an accelerated dynamics which has the virtue of projecting the system into the asymptotic scaling regime. This allows us to revisit critically previous works on the behaviour at large time of the two-time autocorrelation and response functions. We also analyse the case of the two-dimensional system at criticality. A comparison with Glauber dynamics is performed in both dimensionalities, in order to underline the similarities and differences in the phenomenology of the two dynamics.