Direct numerical simulations are used to study mixing and dispersion in decaying stably stratified turbulence from a Lagrangian perspective. The change in density of fluid particles owing to small-scale mixing is extracted from the simulations to provide insight into the mixing process. These changes are driven by temporally and spatially intermittent events that are strongly suppressed as the stratification increases and overturning motions disappear. This occurs for times $Nt \,{>}\, 2\upi$, i.e. after one buoyancy period, where $N$ is the buoyancy frequency. The role of small-scale mixing processes in the density (or buoyancy) flux is analysed. After an initial transient, we find that diapycnal displacements due to mixing dominate the dispersion of fluid particles, even in weak stratification. The relationship between the diapycnal diffusivity and vertical dispersion coefficients is found to be strongly dependent on stratification. Models for the mixing following fluid particles are investigated. The time scale for the density changes due to small-scale mixing is shown to be approximately independent of $N$ and instead remains linked to the energy decay time scale which is relatively insensitive to stratification. There are large changes in the structure of these flows as they evolve under the influence of buoyancy forces. We investigate these changes and their relationship to mixing. We find that strong mixing events are closely linked to the presence of overturning regions in the flow, and that they occur close to (but not within) these regions. The results reported here have implications for the development of improved models of diffusion in stably stratified turbulence.