A shifter pulls or releases a fixed amount of cable each time you hit either of the buttons. That's the cable pull. For each mm of cable pull, the rear derailleur moves a certain amount laterally - the actuation ratio describes how much (e.g. if it's 1.5mm, then each mm of cable pull causes 1.5mm of lateral movement at the RD).
The 9-speed cogs have a larger distance between the cogs than the 10-speed cogs. Thus, if you simply put a 9-speed cassette in the rear, the RD has no way to know that it should move a different amount. Thus, the chain will not shift correctly. There is no way this will work.
The fancy electronic groupsets could, in theory, be programmed for different numbers of rear cogs. In practice, the major manufacturers don't allow this functionality for business reasons. Alternatively, if you have friction shifters, this won't be a problem - however, there are no integrated friction shifter/brake lever setups.
In any case, what is the reason to think about putting a 9s cassette on? If the thought is that 9s chains are more durable, I believe this isn't true; the durability of chains from 9 through 11 speeds has increased with each iteration, probably due to better material quality, better low friction or anti-wear coatings, and probably other factors I don't know.