I'm finding that while BBC Radio 2 and 4 play via my Echo Dot, both stations drop out frequently and eventually just seem to stop playing altogether. This happens all of the time, making BBC stations pretty much useless to me. Classic FM plays just fine.
The odd thing is that if I place my iPhone, which is connected to the same WiFi as the Dot, in the same location and use the TuneIn app on the phone, it streams all BBC stations flawlessly!
My guess is that the Dot is using a different set of TuneIn servers to the iPhone, would love to be able to get to a command prompt and use netstat to check!!