For a start you say One DNS-320 has two 3 TB HDs as RAID-1 for a 6 TB volume.
that is impossible. If you have RAID 1 then you have only 3 TB capacity (the two disks are a mirror of each other), on the other hand if you have RAID 0 then you will get a 6 TB volume.
Now a few questions.
1) what operating system are you using on the computer/s you use to transfer files?
2) have you tried to use FTP to transfer the files?
3) are these large (greater than 1GB) or small files that you are trying to backup?
4) What backup software are you using?
You're correct Ivan. It was a brain fart that made me type RAID 1 instead of RAID 0.
1) I use Windows 8.1 and Windows 10 PCs.
2) The same problem occurs when I use ftp to copy files (from PCs internal or external HD) to a NAS volume.
3) One HD has mostly large files ( 6,000 files, 2.3 TB total). Another HD contains small files (1,100,000 files, 3.0 TB total). Others have a mix. All have the same problem.
4) To copy files I've used ShareCenter Local backup, Comodo Backup, Karen's Replicator, WIndows Explorer file/folder copy, Teracopy, FastCopy and ftp with the same results.
Prior to the DNS=325 and 320 NAS units I used DNS-321 with two 2 TB HDs. I used both one vol per HD in one and RAID 0 in one with no problems.
I keep thinking that there must be something simple that I'm missing and I'll do a facepalm when I find out.