A method and apparatus for migrating one or more data sets each having one or more extents from one or more source logical devices to one or more target logical devices concurrently with interaction between the application and the data being migrated. A background operation copies each extent from the source logical device to the target logical device in a copy state. When a certain level of data has been copied, the extent is locked to assure synchronization of the data in the target logical device to the corresponding data in the source logical device. The status is changed to a mirrored state. When the extents for a data set in a source logical device or in a group of data sets have been mirrored, all the extents are changed to a diverted state. I/O requests to the diverted extents thereafter are intercepted and processed according to whether they access an extent that is in the copy, mirrored, or diverted state.