This is yet another instance of taking the ubiquitous balloon analogy too far.
See, while it's a wonderful way to express the expansion of the universe, there are some misconceptions that arise from it:
- We live in a universe of finite size (we don't know, but we think not) and non-zero curvature (according to WMAP, we don't, or at least we think we don't)
- There are higher dimensions in which our 3+1-dimensional "universe" is embedded (there may very well be, but they aren't necessary for an expanding universe)
- There is something inside the balloon that is causing the expansion.
The latter pops up (in my experience) less often that the others, but it seems that you have hit upon it. It seems that you're treating dark energy as the "something" expanding the balloon.
First off, dark energy is not the cause for the expansion of the universe. When I was younger, I used to think this, but it's absolutely false. The universe can go on its merry way, expanding, without dark energy. But to accelerate that expansion, we do need something else. That something else is dark energy.
Ah, but I need to backtrack. I discussed dark energy, not dark matter. Well, that's because I think you mixed the two up, as anna v so helpfully pointed out. Dark energy causes the acceleration of the expansion of the universe; dark matter causes weird galactic rotation curves. We know that dark matter is in the Milky Way - and in loads of other analogies. So even if you didn't make a typo, the premise is incorrect.
Is dark matter moving? Well, relative to what? There's no absolute reference frame, so we have to clarify what reference frame we're discussing.
If we're referencing
- another galaxy, then the answer is yes, because the Milky Way is moving with respect to the reference frame of that galaxy.
- the stars in the Milky Way, then the answer is also yes.
In fact, if I say that the reference frame is in a galaxy which is receding from us faster than light, then dark matter is indeed moving faster than light. Sort of. But so are we.
And yes, dark matter moves (linked in a comment by lemon).