Your system has some very odd behavior with respect to these SSD's. When moving a file on the same filesystem (often disks correspond to filesystems, but not always) what happens is that OSX (and this is true of Unix systems in general) doesn't actually touch the file itself at all, but just changes directory (folder) entries. An entry pointing to the file is made in the new location, and the entry pointing to the file at the old location is invalidated. The error -43 indicates that the entry at the old location is still showing up in some fashion. The pointer to the file is deleted (which gets you a file not found error), but at least some portion of the old directory entry remains, allowing you to see it in Finder. Doing a "Verify Disk" (not "Verify Disk Permissions") in Disk Utility should find these dangling entries if they actually exist on disk, and aren't due to something (like a third party utility) mucking with the Finder.
Given that this is non-standard behavior, this isn't a problem in stock Mac OSX itself, and there is no standard setting that you can fiddle with to get it to behave correctly. I'll give some possibilities, you can decide for yourself which are most likely and which you want to try.
1) Something has gone wonky with OSX temporarily. Fixable with a reboot. I expect you've tried this.
2) Something has gone wonky with an OSX system file or driver on disk (so it will persist through reboots). Some people will tell you to reset PRAM/NVRAM, but this doesn't sound like a problem fixable with that. You can Google how to do that on your model. A file being corrupted isn't something that Disk Utility or other third party utilities can always detect. Those utilities look for hardware corruption or corruption in the structure of the filesystem itself. If it is simply a change in the file itself, it can be completely invisible. I've had to do this due to losing power unexpectedly and the UPS failing to cut in. It was easier to simply reinstall the OS than to try to figure out which file(s) were corrupted. Incidentally, this happened with an SSD that had been problematic in the past. Early on, some manufacturers simply assumed that SSD's would be installed in Windows computers and based their implementations on that. Given that you say this issue happens on more than one computer, this is probably not the cause of your problem, you wouldn't see the same kind of weird behavior on both.
3) The desktop folder might be corrupted. However, you say this happens with both the Mac Pro and the Mac Mini, so this is unlikely. It would also be picked up by running "Verify Disk" with Disk Utility.
4) Since you say that this also happens on a Mac Mini with SSD, I would actually lay odds that the issue lays with either the SSD's themselves (are they all the same brand?), or a low level third party driver or utility installed on both computers that is interacting with the SSD's in a broken fashion. Or some third party utility that interacts with the Finder is leaving phantom folder entries that don't really exist. The OS code that handles moving files has no clue whether it is interacting with an SSD or HDD, it does the same thing with both. If it is an issue with the SSD making unwarranted assumptions about caching, you would see the behavior with all file moves on the SSD (not just the desktop). If you're seeing it with only the desktop, then something at a higher level (a Finder utility, perhaps) is treating the Desktop differently and producing phantom entries.