Three-level data structures arising from repeated measures on individuals who are clustered within larger units are common in clinical and population health studies. An additional complexity arises when individuals move between clusters over the course of the study resulting in a cross-classified data structure, which needs to be accounted for in the analysis. In these studies, missing data are also a common occurrence. Multiple imputation (MI) is a popular approach for handling missing data, but its validity depends on appropriate tailoring to the target analysis. In the context of cross-classified data, this means that the three-level structure and the time-varying cluster membership should be appropriately accommodated in the imputation model. While three-level data can be handled by either adapting single- and two-level MI approaches using dummy indicators and/or imputing repeated measures in wide format, or using a three-level MI approach, the implementation and comparability of these approaches in the context of cross-classified structures remain unclear. We conducted a simulation study to evaluate MI approaches handling incomplete cross-classified data when the substantive analysis uses a cross-classified random effects model. The simulation design was based on a longitudinal cohort study estimating the effect of depressive symptoms on the academic performance of students over time, clustered by time-varying school. The simulations were conducted under various missing data mechanisms and strengths of cluster correlation. The approaches evaluated included ad-hoc methods, ignoring the time-varying nature of cluster membership by taking the first or the most common cluster; pragmatic extensions of single-level and two-level MI approaches within the joint modelling (JM) and the fully conditional specification (FCS) framework; and a three-level FCS MI approach specifically developed for handling cross-classified data. We also compare the approaches in the longitudinal cohort case study. Simulation results indicated that the FCS implementations performed well in terms of bias and precision while the JM approaches performed poorly. The results in the case study were in line with simulations, with the JM approaches resulting in comparatively different estimates for the variance components than the FCS approaches.