Downloads cannot be made from luminarypodcasts.com, but through the Android and iOS apps, your Downloads found in My Shows is a great place to store episodes you'd like to listen to without cellular service. To add an episode to your downloads, simply tap the ⬇ button (below) when viewing the episode list of a show or the episode's details page:
When the episode is downloaded, the button will turn blue:
If you play an episode from within your Downloads, episodes will continue to play until the end of the list is reached. You can rearrange these episodes by long-pressing on any of the episode cards and dragging to move them.
To remove an episode from your Downloads, tap on the blue download button depicted above and it will delete the download from your device.