This seems like one of those dumb questions, where the answer is, "that happens automatically, silly!"
But as far as I can tell, while my podcast subscription shows the last 10 or so episodes, they don't seem to be saved on the device.
I came to this conclusion when one stopped playing as soon as the buffer ran out after losing signal, and the others all became unplayable (greyed out).
Is there some way to save episodes on-device int he Apple app?
If not, is there an app that can (which also handles subscriptions and works with the device-wide audio UI, so headphone controls work, etc.)?