What noise does a falling missile make?

First lets assume that the action does not take place in vacuum, since then the sound would not propagate as it needs a substance to pass the mechanical wave through.

Second, there is a significant difference between missiles (guided rockets) or unguided rockets and bombs

导弹/火箭有一个电动机(如燃烧solid / liquid motor or a jet engine) which keeps them flying in an aerodynamic regime based on their properties in order to reach the target with certain speed in a certain time and under a certain angle, even for those that work on ballistic principle after the fuel is finished. That means that after they reach the "cruise" speed, it is usually kept until the fuel is out or until the control system decides to turn the motor off. So they usually keep to a certain range of speeds when the motor is on.

Bomb or ballistic rocket is another story. When working under Newton's laws of motion, it will accelerate (when falling). If falling through an infinitely long tunnel filled with air (remember the sound-transmitting substance? ;) ) it will usually quite quickly reach a velocity at which the induced drag forces produced by the air cancel out the Newtonian gravity (acceleration) that pulls it down. This (known as Terminal Velocity) would only lead to a sonic boom if the bomb is aerodynamically designed with Supersonic Terminal Velocity.

So after a while when the fuel is gone and the missile / rocket / bomb reaches its terminal velocity, it will continue to fly with such velocity until its target is hit, or the air becomes so dense it heats up and disintegrates like a meteor.

The sound would not be a whistle, a dropping whistle or a rising whistle but a woosh, growing faster in its amplitude than its doppler shift since the speed is rather constant.

It is true that the whistle was added for emotional distress of the enemy, but also in many cases you would not want your enemy to know that the drop is coming so ...

And, being a videomaker myself, I am ashamed, that knowingly I was using said sounds again and again, and will probably continue to use them as long as my clients demand it :( In this case public pressure and requirement to survive and put food on the table wins over pure science :(