I am using the Smart Keyboard on a 12.9" Pro on 9.3.2. I am having the same issues. I have found that this happens to me, or I should say initiates, mostly when opening a link from one application that opens another. For example clicking a hyperlink in Mail that Opens Safari (you can see the <- Back to Mail in the top left corner. When I go back to Mail then the problem persists. The keyboard commands such as Cmd+Tab also do not work. However when I go back to Home Screen (Using touch) the commands return. Do you see the same?
I suspect this is not a keyboard problem as it appears that iOS displays the recognized commands when you hold down Cmd. Therefore iOS is managing the available inputs. But this is simply a layperson's SWAG.
It would be great to have Apple Support weigh in on the conversation. I think there are a few of us that still find value of leaving our hands on the keyboard.
*** Remembering the Open Apple and Closed Apple key days on my Apple II ... :-D *****