I think you missed the fact that the podcast continues to play when he leaves wifi access. The podcast is not streaming, it has been downloaded already.
I have the same problem. I have podcasts that HAVE been downloaded. I have cellular data turned off, i.e. I am not connected to cellular data. I AM connected to wifi. I start playing a downloaded podcast. I leave wifi access (driving in car). Podcast continues to play. If however I stop the podcast for any reason, I cannot restart it. I get a message that the podcast is unavailable. This just started last week. I've never seen this behavior before. I have an iPhone 5s, running iOS 7 (whatever the last version of that was), i.e. I have not upgraded or made any big changes lately, other than possibly getting the latest version of the podcast app, if one was released recently, maybe last week when my problem started?