In this paper, the authors consider a selection of load-balancing strategies for different adaptive mesh scenarios and in particular for a parallel adaptive multigrid solver that has been developed for the implicit solution of phase-field problems arising from solidification problems. These problems are particularly demanding because they track a moving boundary with a phase field, where a very fine mesh is required locally, and, moreover, evolve large-scale diffusion fields such as temperature where a much coarser mesh may be sufficient. The authors compare a number of standard approaches and a new technique that is proposed specifically for multigrid solvers, where the solver is here based upon a cell-centred finite difference scheme and fully implicit time stepping. They take into account the sequential feature of the grid correction used in multigrid methods. The paper focusses on two phase-field example problems modelling the rapid solidification of an undercooled binary alloy, using isothermal and non-isothermal models, respectively. They observe that the optimal choice of the proposed load-balancing strategy depends strongly on the computational features of the tested problems, namely the computational cost of smoothing and the communication cost of grid transfer operations.