When I am sync'ing a large volume of files, usually over several GB, the target/destination HDD gets stuck at some point. I can't do much with it at that point.
Is it "stuck" no matter how long you let it sit? For example, if you were to let it sit there for an hour, would it eventually finish?
One common cause of confusion is that when you copy files to a drive, Windows will normally try to cache writes to the drive. (I believe it doesn't do this for USB FAT drives by default, but it does for NTFS drives).
So when you start a copy you see the progress bar moving along at a pretty good clip that's limited only by the speed at which it can read the file from the source drive. That can be very fast indeed if the source drive is something like an SSD.
However if the target drive is slower, then at some point if the file is large enough Windows will run out of RAM memory to cache the writes into. At that point the transfer suddenly slows down to the speed of the target drive, and the progress bar seems to stop moving (or moves only very slowly). If the output drive has a very slow connection (i.e., USB 2) then it can get "stuck" this way for a long time as the data trickles out to the drive. It'll be faster for a better connection (i.e., USB 3) but it can still seem quite slow compared to data coming from an SSD, for example.
Attempts to do something else with the target drive (i.e., open an Explorer window to see what files have been copied) can be impacted as well since the I/O requests needed to access the drive get queued up behind the ones already issued for the copy.
If the "stuck" condition you're seeing eventually clears itself if you leave it sit for long enough, then I'd suspect that this is what you're seeing.