A method for migrating volumes in a storage system includes identifying an extent of data (belonging to a volume) requiring migration from a source extent to a target extent. The method allocates a selected number of copiers to the extent of data to migrate the extent of data from the source extent to the target extent. Each copier is configured to copy a unit of data, which is a smaller division of the extent of data. The method monitors destages (i.e., writes) that occur to the source extent as the copiers migrate the extent of data from the source extent to the target extent. In the event the destages occur faster than the copiers can copy units to the target extent, the method allocates additional copiers to the extent of data to assist in migrating the extent of data. A corresponding apparatus and computer program product are also disclosed.