What is the location of an object in expanding and finite space? I was thinking that does it mean the same thing when we say that we are trying to find a stationary object in expanding space or trying to find a moving object in a finite region of space?
 A: You need to be clear about what you mean by "stationary" and "moving".
If an object in expanding space(time) is stationary with respect to the expansion, it will appear to other such "stationary" observers to be moving away at the expansion rate. If that object is the only thing you have experimental access to you wouldn't be able to tell whether the spacetime is expanding or whether the object is just moving away.
However, in expanding spacetime objects which are not stationary in this sense will tend eventually to come to rest with respect to the expansion. That is, over time, objects which are moving away at some rate different from the expansion rate will asymptote towards it. That won't be true in flat spacetime, where any rate of inertial motion is indefinitely sustainable. So if you see very many objects all moving away at a uniform expansion rate, expanding spacetime is a more natural description.
A: In fluid dynamics, this is known as the equivalence between Lagrangian and Eulerian points of view. Given the position function of an object, you can always find a related function to for the movement and/or expansion of medium it is in that would exactly result in apparent movement of a particle and vice-versa.
