I understand that gravity is viewed as flowing as a river pushing objects down on the body of a planet. If that is the case and earth is a sphere, where does the gravity go when it hits the center of the planet? Does it get compressed? Is it destroyed?
I take it you're referring to the river analogy that's sometimes used in connection with Painleve Gullstrand coordinates.
That is often applied to a black hole, where the "river" is a fanciful description of the "flow" of space into the black hole. This flow increases as you approach the horizon, at which point its speed is the speed of light.
You're worried that if we apply this to an object like a planet, where the surface isn't inside its Schwarzschild radius, that the thing that's flowing in the river, i.e space, would somehow "pile up" at the planet surface or center? There is no problem with this as spacetime points are not physical objects, but mathematical abstractions.
The apparent "flow" of space is just a way of describing the way you associate points on a slice of space with points on the slice of space an instant of time later. This flow is the shift vector you get when express your spacetime metric in the form defined in this question.