You’re running into what’s called I/O contention. Duplicating data (simultaneous read/write) on a RAID array is a quick way to bring it to its knees. I had a traditional RAID-5 server array that dropped to <1MB/sec in I/O performance while duplicating. It doesn’t surprise me that the Drobo would suffer the same thing.
What happens is internally the drives seek to one location to read data, then seek to another location to write (seeking is the physical movement of the drive heads to be positioned above the right spot on the platter to access the data). If this is brief (as it is with normal access) it can batch certain operations and avoid any real contention and thrashing. If you’re duplicating data, however, this becomes unavoidable. Since seeking takes much more time than the actual reading and writing of the data and you’re seeking constantly, you spend most of your time moving the heads across the disk rather than reading/writing data, and performance plummets. Add to that all of the housekeeping the DroboFS has to do, and while I don’t know if it hurts, it certainly can’t help matters.
As to your current problem, if you need to move data between shares it’s MUCH easier and faster to SSH into the DroboFS and just move the data. Shares are an illusion - on the underlying filesystem they’re just different folders, so you can move data between them without copying - instantaneous.