You will find that gravity behaves exactly like a low pressure system. That could be a low pressure system in the weather, where air (and blowing leaves) are sucked into the centre of the low. Or low pressure in water, where a cork will shoot from the bottom of a lake to the top. So the motion of something falling is just that object following the pressure gradient from high pressure to low pressure.
But what is the pressure in? I would conjecture that it is a low pressure of time. That is, time runs slower near a planet, so all objects want to move where time runs slowest.
All objects fall at the same speed, from a feather to an anvil, because falling has nothing to do with the object itself. It only has to do with the pressure above and below the object.
So don't think about the standard bowling ball in a trampoline model of warped spacetime. Think more about a weather pressure map where the wind is sucked into the middle. This is what causes objects to fall.