Is Dark Matter moving? It sounds like dark matter makes up one quarter of the universe's mass-energy.  And it's density is fairly uniform.  But could it be simultaneously both uniform and moving - like currents in a lake.  Does it have a standing wave?  A propagating wave?  A rotation around some universal central point?  Local rotations around massive objects like low pressure systems on a weather map?
Yes and no. In general, Dark Matter tends to exist in the same places as galaxies. It isn't really distributed throughout all of space uniformly. It has voids and filaments just like normal matter does. In fact, on the whole, normal matter is a pretty good tracer of where Dark Matter is. In some cases Dark Matter can separate from normal matter, and it's believed by some that Dark Matter can clump in areas where normal matter doesn't, but more or less, on a cosmological scale, it is distributed and acts like regular matter. On short timescales, Dark Matter most just sits there. There isn't mass motion or propagating waves. On long timescales and in very large views, you can see actual motion and interesting structures (e.g., Baryonic Acoustic Oscillations - technically we observe these in normal matter, but Dark Matter would be affected in the same way).

To give you an idea of this, check out the Millennium Simulation (that's a YouTube video which moves around and views the results of the simulation). This was a set of simulations of the universe as a whole, in an attempt to test out current theories about our universe and see if those theories could produce a universe that looks like ours. Suffice to say, they were pretty successful. An image of the results of the simulation is shown below. This represents the universe at a very very large scale and you can see all the dark matter and how it is distributed.



You might also want to check out the Aquarius Simulation which was very similar in nature to the Millennium Simulation. However, unlike the video for the Millennium Simulation, this video shows the evolution of the universe. This really gives you an idea of the flow and motion of dark matter over the entire history of the Universe.