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.