I am using the Web AppBuilder Developer Edition 2.0 with the Local Layer Widget.
I set up my basemaps in the Enhanced Basemap Widget. However, after I create the app, download it, and unzip it to my inetpub/wwwroot/webmaps folder, the app opens fine. But, when I open the Enhanced Basemap Widget, and click on a basemap (I added 3, one is hosted on AGOL as a Web Map with the service using a local URL, the others are hosted local on my server machine, not AGOL I added them inside the Enhanced Basemap Widget), the basemaps open, then disappear in the widget window after I switch to the other basemaps. Eventually, I only end up with one basemap that is visible.