How to make a warp bubble moving? In answer to this question it is stated that:

In summary, warp drives move because they already move, or else they need to propel, and their movement is not related to spacetime expansion or contraction.

This means that firstly a warp bubble had to be accelerated at some speed. The speed is higher than speed of light (this is a reason why we try to construct a warp bubble). However, how is it possible to break "light barrier"? Is it thanks to a "special" spacetime geometry of the warp bubble that allows a conventional propulsion system to break the barrier?
 A: Currently, there is no known mechanism to getting the warp bubble moving at all, let alone to superluminal velocities. In Alcubierre's original metric, he assumed that the warp bubble would already be moving (see original paper), and didn't consider how that warp bubble would start moving in the first place.
And firing conventional rocket engines or any other propulsion system in a warp bubble doesn't do anything to this effect either. The "inside" of a warp bubble is flat spacetime, and so firing rockets inside of it would be essentially like firing rockets in interstellar space; it doesn't affect the warp bubble itself.
In fact, this is probably the big issue affecting all warp drives that still remains unresolved. Here is another physics SE post that elaborates more on this issue.
A: At the heart of the warp drive debate: what is the physical process that causes the warp drive bubble to move? And what would make it reach FTL speed? As I see it, there are two positions/hypotheses:
A): the original Alcubierre one, which says that the expansion/contraction of spacetime is the causal effect. The direction of travel is that space contracts in front of the bubble, expands behind it. In effect, the mechanism causing the expansion of the Universe "works" locally and in a narrow direction. Just as the c-barrier does not apply to the expansion of the Universe, it does not apply to this. There is no detailed, computational model of how the bubble would reach FTL speed. But this idea does not require it to happen by some conventional spacecraft propulsion, but somehow by expansion/contraction. (Alcubierre paper: https://arxiv.org/abs/gr-qc/0009013), (Alcubierre, Lobo paper https://arxiv.org/abs/2103.05610), (Lobo, Visser paper: https://arxiv.org/abs/gr-qc/0406083)
B) The other one, which has become more common in recent years, is that it is not caused by the expansion/contraction of spacetime, it is just incidentally related to it in some models. There are warp spacetimes with no expansion and contraction. According to this there is no real physical process behind it all, so the expansion of the universe is not a good analogy. So warp drive is, for now, just a form of motion defined on paper that fits within the rules of general relativity. Some unknown and as yet unimaginable engine should provide the FTL speed of the bubble. According to this, the position of the warp drive would in fact be the same, not more realistic, than that of the imaginary tachyons (Bobrick,Martire paper: https://arxiv.org/abs/2102.06824)
To our knowledge, neither hypothesis A) nor B) are proven or disproven, but are competing alternatives. An important question that arises here is: if a new warp drive model is discovered, such as Lentz, which version does it fit? Can we place it among those in which expansion and contraction can be detected, and thus establish that such a realistic effect can generate FTL motion? Or, on the contrary, is it rather a counterexample?
However, the proponents of version B) claim not only that they have developed a new hypothesis, but also that version A) is a fallacy that they have disproved, and version B) has been proven. This is the point of the Bobrick-Martire study, among many other developments.
But it is not clear where this follows from, where would it be proven to be so? The Alcubierre-warp-drive papers clearly attribute it to the expansion/contraction effect, also to the equivalent spacetime (van den Broeck spacetime etc.) But the main claim of the Bobrick's is that "warp drives move because they already move, or else they need to propel, and their movement is not related to spacetime expansion or contraction". And so you would need some unknown, ordinary drive in addition, which could accelerate the bubble above c. Therefore, the ontological status of the warp drive is, according to them, no more than, for example, the imaginary tachyons in special relativity.
However, I do not see where their position would be proven. Neither in Natario's paper nor in Bobrick-Martire's paper have I noticed such a proof. They do address it, but it seems to be more of a hypothesis, an argument, not a rigorous proof. Thus the question: how is it possible to break "light barrier"? Is it due to spacetime deformations or would it need additional conventional propulsion to accelerate it above c - I think it remains open. (Natario paper: https://arxiv.org/abs/gr-qc/0110086)
A: I don’t think @JS4137’s answer is quite complete. The warp bubble moves because its (fictional exotic) mass moves. The energy-momentum tensor of the alcubierre drive has forward-pointing momentum on the inside of the torus and backward-pointing momentum on the outside of the torus. I’ve seen someone elsewhere on a forum liken this to a smoke ring. It kind of is producing a frame dragging effect on the bubble, pushing it along. I’m currently working on an extension of Einsteinpy that demonstrates this nicely and will share my results in the documentation there.
