Which effect causes a moving source's pitch to drop as it passes you?

The Doppler effect. Waves ahead of a moving source are bunched up and those behind are stretched, so an approaching siren is higher and a receding one lower - and the drop happens at the moment of passing, not gradually.

Buys Ballot tested it in 1845 in the most literal way available: he put a group of trumpeters on an open railway carriage, ran them past a platform of musicians with perfect pitch, and had both parties write down what they heard. It worked.

The same effect on light is how nearly everything in astronomy is measured. A receding galaxy's spectral lines shift towards red, and Hubble's finding that almost everything is redshifted in proportion to its distance is the observational basis of the expanding universe. It also detects planets - a star tugged by an unseen companion wobbles, and the wobble shows up as a periodic shift of a metre or so per second in its spectrum - and it is what a police radar gun and a weather radar are doing.