Unless the sand is enclosed within the 'timer' in a vacuum, the "free fall" of the sand makes no difference. It's the old story of "does a truck full of pigeons weigh less if all the pigeons were made to fly". Answer? No. The pigeons (and the grains of sand) exert their weight on the air which transmits the weight (more accurately, mass) to the bottom of the timer.
However, II_billym is on to something and its called the inverse square law.. Simply stated, if the distance between two objects were to increase by a factor of 10, the gravitational influence exerted by each object upon the other would decrease by a factor of 100. So, the variable here is the small distance between the volume of sand that was originally at rest nearer the center of the Earth's mass is now a minute distance further away... hence an incalcuable (well, not really) small loss of mass until it drains back to the original, lower position...