Placing Swamp Cooler in garage and venting through flooring space; Bad idea?

[orangele]

I live in Las Vegas with very hot temperatures and very low humidity, which is an ideal location for swamp coolers. I have swamp cooler in my upstairs kitchen window which works great. I now am considering placing a swamp cooler in the wall of my garage and placing a vent for air in the garage ceiling space which is the floor level for the second level living room. i.e. the air from the evaporative cooler blowing into the garage would vent into the garage ceiling, and flow in this space to vents which could be placed on the same side of the house as where the cooler will be placed. Please see my very rough diagram of plan. I dont see any reason why this should not work well, but wonder if others have any thoughts why this would not be a good idea.