How object's movement is separated from the space-time movement? In Issues related to expansion of the universe while space-time itself is moving how it is possible to calculate the speed of an moving object? In other words how object's movement is separated from the space-time movement?
 A: The best way to think about this question is to view space expansion as a global phenomenon while relative movement of objects is a local phenomenon.  The only reason we are aware of space expansion is the distance related red shift of radiation received from very distant objects.  By very distant objects I mean objects in galactic superclusters that are not part of our own local supercluster. A supercluster is defined as the largest collection of galaxies that are bound by their gravitational attraction.
Measurements of relative speed of objects in our local supercluster are not significantly influenced by the expansion of the universe because at this local scale the expansion during any small interval of time is very small compared to the dynamics associated movement between objects.
A: Does it matter? Let's say you make a manifold and you friend makes a manifold. And you each put some worldlines on your manifold. And you each put a metric tensor on your manifold. And you each use the metric to compute proper time along the worldlines.
And you mark equally spaced portions of the worldlines (equal according to proper time, using the metric along the curve) and you send out null geodesics from those marks to the other worldline.
And you compare how many null curves hit the one world line from the other worldline compared to how many of the worldline's own markings are along the worldline. You can call that a kinematic redshift or a gravitational redshift because you that's just name calling.
The science happened when you numerically compared how many null curves hit the curve compare to the proper time markings along the curve.
You could do it in your manifold. Your friend could do it in their manifold. If you get the same numbers you make the same predictions so what does it matter what you call it.
This isn't me being facetious because you literally asked how to tell something that didn't affect the calculations, didn't affect the predictions. So it doesn't have an observational answer. There is no empirical answer, which says something about the question.
Basically, coordinates are like a gauge. If you pick one, then that picking doesn't make it physical. And the predictions end up not depending on the gauge or the coordinates, so you don't want to get too excited about the coordinates or the gauge.
So maybe in one coordinate system it looks like a kinematical redshift. Maybe in another coordinate system it doesn't look kinematic in origin. Don't get overly excited by things that don't affect your predictions. Don't shy away either if it helps you get stuff done. But it just is what it is.
