You need to be careful about statements like spacetime is contracting or expanding or indeed doing anything else. Spacetime isn't a thing. It is a mathematical object that we use to describe the motion of things. So it is meaningless to ask whether spacetime contracts.
However what we can do is take a sphere of test particles and see how it changes as it moves through spacetime.
Imagine taking a large number of particles that are too small to exert any significant gravitational force on each other and arranging them in a sphere. If these particles are floating in space far from any other matter then they will just stay as a sphere - the radius and volume of the sphere won't change. But if we now let our sphere move into some gravitational field then it will change in shape and/or volume. So while it doesn't make sense to ask if spacetime expands or contracts it does make sense to ask if our sphere expands or contracts and how that sphere changes does tell us about the curvature of the spacetime.
It turns out that the volume of the sphere is related to an important properties of the spacetime geometry called the Ricci tensor and Ricci scalar aka scalar curvature. Basically an increasingly positive Ricci scalar means the sphere is contracting and an increasing negative Ricci scalar means the sphere is expanding.
The shape of the sphere is related to another property called the Weyl tensor. This tells about tidal forces acting on our sphere.
To make this concrete take your example of the sphere falling towards a massive body - possibly a black hole or possibly just any massive object. When we calculate the spacetime curvature we find that the Ricci tensor and scalar are both zero so the ball stays the same volume. It neither contracts nor expands. However the Weyl tensor is not zero so the ball experiences tidal forces. In fact the ball gets stretched along the line towards the massive object and compressed at right angles to this line. This process is what s commonly known as spaghettification.
So for a massive object if we were going ignore the imprecision and use the metaphor of spacetime spacetime expanding or contracting we would have to say that spacetime is neither expanding nor contracting near the massive object, but it is changing shape.
There are examples where the volume of our sphere does change and the most obvious is an expanding universe. In an expanding universe like ours we find the volume of the sphere increases with time, which is why we talk about the universe expanding. In a hypothetical contracting universe we would find the volume of the sphere decreases with time.
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The interior of the bubble is causally disconnected. It's not possible for the bubble to be turned off or steered from the inside. But there is no reason it cannot be affected from an outside agency at a pre-planned points, or even simply have a finite lifetime, naturally deteriorating to stop at the intended destination.
That is not the case. The statement "you never moved through space" just means you never move relative to the bubble interior, which is different from the bubble itself moving through space. Even if the bubble accelerates, you will not experience any g-forces.
Intuitively, one can think of it as dual to the cosmic expansion of space: as space expands, it carries galaxies along with it, and because it's space between them that's expanding rather than them moving in space, distant galaxies can have superluminal separation velocity. Effective movement because of the way space expands (or contracts) is different from movement in space.
Even if cosmic expansion could somehow be "turned off", it wouldn't suddenly make all the galaxies contract together again. It would simply stop further separation.
The Alcubierre drive does something similar: instead of expanding space to get away from from distant objects, it contracts it in order to approach them. It doesn't actually need to contract all of space in front of it; it just expands it back after traversing it. In effect, the warp bubble rides its own gravitational field.
Although Newtonian analogies are fraught with peril (esp. here, since not only is gravity different, but the behavior of negative mass especially so~don't take this seriously!), but a fun little toy exercise is to consider what would happen if you lived in a Newtonian universe and had two masses, equal in magnitude but one positive and the other negative.