I use Spotify at the moment (thinking of switching back to Antennapod due to Spotify being buggy as fuck) and actually hit download on every episode I want to listen to: putting them in my downloads is the easiest way to both mark them as 'to be listened to' as well as having them available whenever (though that's one of the bugs: I'll still need internet to get permission from Spotify HQ to start playing, but I won't have to load the audio stream and can turn off internet once it's playing).
Or you could just grab the files and dump them into a folder for something like smart audiobook player -- that's what I'd do for anything longer than ~6 hours in total, don't want to keep track of that stuff in my podcast player which is for, well, podcasts, not (cut-up) audio books.
Every time I hear that (and I want to check it out), I do so in Pocket Casts, my fav app (first on Android, then on iOS). I know there are probably podcasts that are _only_ on Spotify, but I haven't tripped over one yet.