Since particles that spin don't follow geodesics and particles with mass or energy or momentum or stress don't follow geodesics, the linked questions and answers about geodesics don't answer your question.
You want to know how real objects move, not how test particles move.
And this is handled by the Cauchy problem in General Relativity. Basically, you specify a 3d space that has some 3d spatial curvature, and you also specify some additional structure that indicates, not the full spacetime curvature, but just a little but about how it is changing right now.
It's like specifying both the initial position and the initial velocity, it's hopeless to get the motion without first specifying both. And just like you can use $\vec F=m\vec a$ to hope to get motion from the initial position, initial velocity, and the force, so you can use Einstein's Equation to try to get the full spacetime.
The Einstein Equation has a left hand side that has the Einstein Tensor. And that Einstein tensor has second derivatives of the metric tensor. And the Einstein Equation has a right hand side that is the Stress-Energy tensor.
If you know the Stress-Energy tensor and you know the metric and the first derivatives of the metric then you can hope to solve for the second derivatives of the metric and then find the full evolution over time.
It's not as simple as in Newton's theory. The first complication is similar to electromagnetism. In electromagnetism you can use the Maxwell Equation to find how he initial electromagnetic field evolves into the electromagnetic field in all of spacetime, but the initial electromagnetic field can not be arbitrary. The divergence of the initial magnetic part of the electromagnetic field has to be zero. So you can't write down just any fields and then evolve them.
Similarly, you can't write down just any initial manifold and any initial Stress-Energy tensor and any initial spatial metric and any arbitrary information about first derivatives.
And secondly, even when you have good initial conditions that meet the constraints. You have to extract the second derivatives (which isn't easy becasue the equations are not linear) and then use them to make a complete solution. Which is also hard.
And furthermore, not every manifold that satisfies the Einstein Equation has a Cauchy problem setup. There are even solutions to Einstein's Equation that have spatial slices that are perfectly flat, perfectly empty, and perfectly infinite and look just like a spatial slice of empty flat Minkowksi space. And yet some of those solutions are different than each other.
So even the simplest possible case of a flat empty infinite universe with nothing in right now, could evolve differently, and there simply is no law of physics telling it that it has to have a particular future development or a particular past development.
Physics is a science. And the theories tell you to restrict your models to only certain models. Electromagnetism tells us to restrict our initial magnetic field to have zero divergence and it tells us to restrict our electromagnetic fields to have their eceokve and their initial values to related to each other in a particular way. So it says to only use certain fields for your models.
And you use the models to make your predictions. So part of the theory is that some particular one of the restricted initial magnetic fields will be able to match your observations, and that one of its evolutions will match your later observations.
So in General Relativity the equation and the theory really tell you to select manifolds and metrics and Stress-Energy tensors that satisfy the Einstein Equation and the evolution equations for the specific things that make up the Stress-Energy tensor (which then cause the evolution of the Stress-Energy tensor).
So the theory says to restrict yourself to just those models. And the theory is good if there are models that match observations.
Think of science as tell you what the world does not do. That's what it has to do, becasue we want science to be falsifiable.
So science hypothesizes that the universe doesn't evolve in a way to look like a manifold that does not satisfy the Einstein Equation. So if it ever did, we'd know the theory is wrong and we could modify it or replace it. It doesn't really do anything more than that.
