UPDATE: As of 2021, GEO Jobe has transitioned from recommending Admin Tools for ArcGIS to Backup My Org when it comes to migrating items, users, and groups from one organization to another.
Generally, hosted services that are utilized by a web map or a web app will automatically be migrated and wired appropriately during a cloning operation. However, if services are registered to a different data source (such as a federated server or, for example a production environment), the item will migrate to the destination, but the URLs will still remain pointing to the original source, and the references will need to be updated to the new location where the services exist. Admin Tools does have the capability to perform this task with the "Update Items Content URL” tool:
Additionally, there is a dedicated in-app Item data editor to update the URLs individually. This editor is accessed via drop-down option for the item in question in the filterable list view in any of the tools:
NOTE: For Web Maps, you may also need to employ the “Find and Replace Web Maps Services URLs” tool.