GRT backup is starting well, but after "some time" is slowing down .
When you said this, did the job actually start reading or writing any data to storage? Check the byte counts?
If it has started writing, the socket failure in the logs may not be relevant.
Check bptm logs on media server, does it tell you anything about "Kbytes transferred", see how fast/slow it is over time.
If farm backup (non-GRT) is fine, I don't see why GRT will be a "connection issue", ensure nbfsd are running fine on both media server and clients (frontend + backend), check nbfsd logs for more details