Summary: We consider the low-temperature coarsening dynamics of a one-dimensional Ising ferromagnet with conserved Kawasaki-like dynamics in the domain representation. Domains diffuse with size-dependent diffusion constant, \(D(l) \propto l^\gamma\) with \(\gamma = -1\). We generalize this model to arbitrary \(\gamma\), and derive an expression for the domain density, \(N(t)\sim t^{-\phi}\) with \(\phi = 1/(2 -\gamma)\), using a scaling argument. We also investigate numerically the persistence exponent \(\theta\) characterizing the power-law decay of the number, \(N_p(t)\), of persistent (unflipped) spins at time \(t\), and find \(N_p(t)\sim t-\theta\) where \(\theta\) depends on \(\gamma\). We show how the results for \(\phi\) and \(\theta\) are related to similar calculations in diffusion-limited cluster-cluster aggregation (DLCA) where clusters with size-dependent diffusion constant diffuse through an immobile ‘empty’ phase and aggregate irreversibly on impact. Simulations show that, while phgr is the same in both models, \(\theta\) is different except for \(\gamma = 0\). We also investigate models that interpolate between symmetric domain diffusion and DLCA.