Quote from my friend:
I had a number of podcasts saved in the Apple podcast app, and it turns out that the content for that app is not included in iCloud backups—so the content didn't come over to the new phone via restore. The next discovery is that the Apple podcast app also doesn't have a “share” function, so no way to export the content that way. I then installed iTunes on my PC (malware), and iTunes lets you see the content nicely listed—but again no way to copy to the new phone or even extract to PC. Finally I got a third party software—more malware, and full of bugs—but finally was able to save the content to the PC. Then I found a podcast app that supports importing media files (it is called Downcast), transferred from the PC to the new phone, imported to that new podcast app—and done. This took hours, in part because once extracted to PC from the old phone, the meta-data about these files (order, when received, etc.) was lost, so I had to rename the files to preserve the correct order.