I spoke a bit too soon. It is possible to send stuff to DropBox and other services, but they hid it deap. You have to chosoe the action icon next to the file name (box with up arrow), tap the Share item, then Send Attachment, then Send Copy.
At that point you will get the normal share sheet. Besides emailing the attachement you can choose to save a copy to DropBox, Gmail, or other compatible cloud services.
I'm glad you asked this question, because up to now I've been cussing at Word for not letting me move files back out of OneDrive. Awkward as it is, at least it's an option.