It's strange to me that I can't find anyone with the same issue, because it's happening to me in two different though similar environments...
I am creating FileTables and FileShares with non-transacted access "Full". I have a problem with several scenarios, but I'll detail one of them:
Scenario:
UserA copies a file to the FileShare (via mapped drive X:). File is visible in the share from the Server and UserA's computer. UserB doesn't see the file (also mapped drive). I move the file while logged in to the server to another folder in the FileShare of the same FileTable. The file still appears to be in the original location to UserA for a period of time (UserA can even open it), but then it appears to be gone from the original location. However, it doesn't appear in the new location to UserA. Meanwhile, UserB never sees the file.
Now UserC logs into a remote desktop. UserC sees file in new location. UserA copies a new file to the FileShare. UserC can't see the new file. UserC logs off remote desktop and logs in again. UserC sees the new file.
UserA connects to the same location using a different name in DNS and mapped drive Y: and sees the file in the new location. On drive X:, file still isn't visible.
Bottom line - A remote connection to the server always shows files in their correct folders. Clients, however, can't see actions taken by other users or from the server unless they log off their desktop and log in again or connect to the server's FileShare using a different DNS name.
This all seems pretty crazy to me, but there has to be a reason why I'm seeing this behavior consistently in two environments.