Efficient and distributed adaptive mesh construction and editing pose several challenges, including selecting the appropriate distributed data structure, choosing strategies for distributing computational load, and managing inter-processor communication. Distributed Combinatorial Maps permit the representation and editing of distributed 3D meshes. This paper addresses computation load and expands communication aspects through volume transfer operation and repartitioning strategies. This work is the first one defining such transfer for cells of any topology. We demonstrate the benefits of our method by presenting a parallel adaptive hexahedral subdivision operation, involving fully generic volumes, in a process including a conversion to conformal mesh and surface fitting. Our experiments compare different strategies using multithreading and MPI implementations to highlight the benefits of volume transfer. Special attention has been paid to generic aspects and adaptability of the framework.