iTunes 12.4.3 (on OS X 10.11.6) : podcasts : issues with HTTP/1.1 redirect 302?
The problems turns out to come from my ESET Endpoint Antivirus 6.3.85 (and earlier) on OSX 10.11.6 (and earlier). More particularly ESET's Web Access Protection.

When I disable it, everything works fine. 
To recap, the problem appears with any software accessing URLs like 

http://rf.proxycast.org/1181578500855832576/14489-02.07.2016-ITEMA_21024836-0.mp3 (1)

That URL redirects to

http://media.radiofrance-podcast.net/podcast09/14489-02.07.2016-ITEMA_21024836-0.mp3 (2)

URL (1) actually breaks in iTunes, Safari, Chrome, Firefox.

URL (2) works fine anywhere.

I tried excluding the domain "proxycast.org" from the URL scanning list, but that does not solve the problem completely.
iTunes still fails to handle correctly the redirect it seems, as does Safari. Chrome and Firefox do play the URL.

Right now the only, temporarily acceptable workaround I can find is to exclude the IP address of (1) scanning by ESET.
This will only work of course until that IP changes.

So to conclude:

There is an issue linking HTTP redirects, ESET Endpoint Antivirus for Mac OS X, and iTunes for OSX (or Safari). 

PS: I don't have this issue with those podcasts on other machines, Mac and PC, that do not run ESET.