But the chances of this ever happening are extremely low (as in, lots of zeros after the decimal point low). An old end-of-earth fantasy book I read as a child got it wrong: the Moon is not falling, and is in fact currently receding from us at roughly 4 cm/year. (That rate is not constant.)
The physics are complicated. (Very complicated.) But we can outline the basics by talking about tidal forces. The Moon exerts a gravitational pull on the Earth, squishing the Earth to slightly more like a football than a sphere, an oblate spheroid. Earth's oceans can deform more to this pull than land, but because the Earth spins more rapidly than the Moon revolves, the bulge of the oceans is whirled slightly ahead of Earth-Moon shortest distance line. Crash Course Astronomy has a good animation at about 5:00.
But this bulge has mass, and so it exerts a gravitational pull on the Moon, hurrying the Moon along. The Moon also pulls back on the bulge, which drags on the Earth and slows down its rotation.
The net effect is to transfer energy to the Moon, causing it to speed up, and because of Kepler's Second Law (orbiting bodies sweep out equal areas in equal time, seen better with an animation), this moves the Moon farther away from the Earth.
There are lots of complicating factors: the shape of the continents (not constant), friction between the Earth and its oceans (imagine trying to model tidal forces during a global ice age!), the forces exerted on and by Earth's atmosphere, every other non-negligible body in the solar system exerting its own gravitational force into the mix, orbital mechanics, etc.
Ask an Astronomer has a nice write-up without math. I don't have a good source for those wanting math, but I'm starting with Lunar Orbital Evolution: A Synthesis of Recent Results from 1999 (I know, not that recent, but free.) If you, gentle reader, have a relevant article that you found helpful, please leave the title and a link in the comments.
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