Which mobile platform are you using? iOS, Android...?
Have you downloaded the mobile app or are you using the native web browser on the mobile device?
Which sample are you using? The link does not go to a specific sample.
Are you saying your web map displays fine in the browser with your custom basemap, but not when you attempt to view it on your mobile device?
Perhaps the easiest thing to do is to share your web map publicly so I can look at it.
Thanks,
Mike