What type of custom basemaps you have created, tiled or dynamic maps? An easy way to consume your own maps/layers is to add them as a webmap/featurelayer to your ArcGIS Online account. Once they are available on ArcGIS Online, you can add them to app's compilation panel via their URLs.
I’ve tried with tile layers. How could I convert them to a feature layer?
Brian Landes | TRANSWESTERN | Director of GIS/Location Intelligence | Direct: 925.357.2021 | Mobile: 480.294.7622
If you basemap is a dynamic map service, the app automatically adds all layers it contains as vector feature layers once you add this dynamic service layer into the Compilation window. Tiled map service is a little complicated. If it is not tilesOnly type of the tiled map service and has access to its data, our app treats it the same way as dynamic service and load all the feature layers it contains. Otherwise, it is loaded as a raster layer.