On the Amen iOS app, meditations, bedtime stories, and the Rosary can be downloaded for future listening offline.
To download a meditation, bedtime story, or the Rosary on an iOS device:
Start the meditation, bedtime story, or the Rosary for the time length you'd like to listen.
You will see a download icon at the upper right corner of the track player screen, see picture.
Tap that to download the current track you are listening to.
You will find your downloaded meditations under your Profile tab.