Can you please elaborate some more on this?
Is this a mechanical thing?
Definitely cannot use the DQS to downshift. The concept of the DQS is this:
When you are on the throttle, the engine is putting pressure on the transmission, which in turn is translating it to the wheel to move you forward. When the trans has stress on it, it's difficult/impossible to pull out of a gear and move into another. Thus why we have a clutch, to relieve the pressure on the trans. The DQS cuts the throttle at the moment you push the shifter just long enough to relieve that stress and allow the trans to switch gears.
Downshifting is different yet the same. Obviously you are not on the throttle, but the engine is still putting stress on the trans by way of resistance. The engine wants to slow down and you are forcing it to go faster by switching into a lower gear. It is very possibly and mechanically safe to switch into a lower gear without using the clutch. If you are on the throttle, the moment you let off of it, you can click down into a lower gear, because at that moment is when the change of forces happens between the engine and trans (i.e. engine pushing trans verses trans pushing engine). It's also possible, when you are slowing down to blip the throttle just the tiniest amount (again creating this change in forces) and click down into a lower gear at the exact same time. This is harder and probably not as healthy for your trans unless you get really good at it (it took me a while on my last bike, but I got really good at it).
In reality, the absolutely best way and practice is to use the clutch, blip the throttle, and drop down a gear. This will give you the lowest risk of damaging anything, as well as stabilize the rear tire, especially if you are pushing it on a track. It's a very good skill to develop and I highly suggest you practice it while coming to stops.
You can still downshift the other two ways if you like. Chances are you won't really damage anything unless you are just horrible at it and do it wrong constantly.