Is Dark Matter in Motion? What is known about the motion of dark matter, especially in galaxies?
It seems as though a particular distribution of dark matter might be required to cause the very flat galactic rotation curves such as the one for the spiral galaxy NGC 3198 below.

Have models been investigated for different motion e.g. where the dark matter orbits at constant radius, or is stationary, or is even moving at constant speed towards the galactic nucleus, to be periodically ejected?
Do any of these models account for the flat rotation curves that we associate with visible matter?  That is, which models for dark matter's motion best lead to the distribution of it that's necessary to cause the flat rotation curves?
It would be interesting to hear about any work that has been done on this.
The question is motivated by the investigation of a model that has a constant velocity in-fall of dark matter directly towards the nucleus, that then meets a constant density region.
The points (0,0), (1,8), (2,8), (3,8) etc... were plotted with a cubic spline fitter to see what would happen and the result appears quite like a rotation curve.

 A: The simplest model for dark matter motion is virialized random velocities. This is a reasonable conceptual description for dark matter motions in simulations as well. Rather than simple orbits, think beehive or cloud of particles. A chaotic system of many particles interacting gravitationally and exchanging energy is as messy as it gets, but luckily, messy as it gets can sometimes be described well with statistical mechanics.
Something essential you want to learn about is the isothermal sphere, wherein self-gravitating particles with a "thermal" distribution of energies naturally produce $\rho(r)\propto r^{-2}$ density profiles and hence flat rotation curves. Some other things you might like to learn about are the collision-less Boltzmann equation, describing distributions of particles and the Poisson equation, relating density to gravitational potential.
Isothermal Sphere
I adapt a simplified argument from Ch 4.3 of Binney/Tremaine's Galactic Dynamics.
Imagine the dark matter has a Maxwell-Boltzmann distribution of energy-per-mass $E = \frac{1}{2}v^{2} - \Psi$ (kinetic and potential energy) for something analogous to a "temperature" T: $f(E) \propto e^{-E/T} \propto e^{(\Psi - \frac{1}{2}v^{2})/T}$.
Integrating the 6-d phase space distribution f(E) over velocity space $d^{3}v$ leads to a density distribution $\rho \propto e^{\Psi/T}$ or $\Psi = T \ln \rho$. Let's ansatz that $\rho = r^{\alpha}$.
Plug this into Poisson's equation with spherical symmetry:
$4\pi{}G\rho = \nabla^{2}\Psi = \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\Psi}{dr}\right) = \frac{T}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\ln (\rho)}{dr}\right) = T\alpha \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\ln r}{dr}\right) =  T\alpha \frac{1}{r^{2}}\frac{d}{dr}\left(r\right) = \frac{T\alpha}{r^{2}} $.
We now see that, because $\frac{d}{dr}\left(r^{2}\frac{d\ln r}{dr}\right) = 1$, we have found that $\rho \propto r^{-2}$. This arose from our assumption of a thermal distribution of energies for our dark matter, which lead to a relationship between density and energy and hence density and potential. We then used another relationship between density and potential, the Poisson equation, to show that the density distribution leading to flat rotation curves is a natural result of a reasonable set of (over)-simplified assumptions.
Flat Rotation Curve
This is well-covered by rfl's answer so I only include this as a footnote for completeness. Given a density $\rho = \rho_{0}r_{0}^{2} r^{-2}$, the mass of a shell of width $dr$ will be $\rho 4\pi r^{2}dr = 4\pi \rho_{0}r_{0}^{2} dr$. So the total enclosed mass within radius $r$ will be:
$M(r) = 4\pi \rho_{0}r_{0}^{2} r$
which leads to a rotation velocity
$\frac{v^{2}}{r} = \frac{GM}{r^{2}} \rightarrow v = \sqrt{GM/r} = \sqrt{4\pi{}G \rho_{0}r_{0}^{2}}$
which is indeed flat (because it is constant).
A: IF dark matter is made of primordial black holes (and this possibility is not excluded by observations, and is even studied intensively by some physicists, see for example: https://www.pbs.org/newshour/science/primordial-black-holes-could-explain-dark-matter-galaxy-growth-and-more ), then it has to be moving.
If the don't merge they can form non-dissipative structures (well, they would emit gravitational waves, but only BH mergers will emit a substantial amount). So they won't show the typical behavior of galaxy formations, i.e, the flattening). They will keep their halo-like distribution, thereby providing an extra inward pull on the stars constituting a galaxy and clusters of them).
Observations on the Bullet Cluster rule out theories claiming that DM is tight to normal matter (as in Verlinde's emergent gravity, where DM is a kind of reaction coupled to normal matter distribution and linked information on a surface enclosing the normal matter), as DM concentrations are seen to exist on their own. So it is matter indeed. The question the remains how huge collections of small primordial BH will evolve in time and how their initial states looked. IF they were there in the first place.
A: Yes, dark matter has to be in motion, otherwise it would fall in to the galactic center. From the fact that galaxies are stable, we can expect the Virial Theorem to hold, i.e. that dark matter has a total kinetic energy of half the total gravitational potential of the galaxy.
Yes, indeed, a particular density distribution is required to result in the observed rotation curve; taken the other way around, measurements of galactic rotation curves are measurements of the dark matter density profile. For simplicity (which turns out to be a good approximation) let's assume a spherically symmetric distribution and equate centripetal and gravitational forces:
\begin{equation}
\frac{mv^2}{r}= \frac{GM(r)m}{r^2}
\end{equation}
with $M(r)=\int \varrho(r) 4\pi r^2 \mathrm{d}r$ the dark matter mass profile. Sanity check: Outside the mass distribution, $M(r)$ can be approximated as a point mass $M$ at $r=0$ and one recovers Kepler's law $v(r)\propto 1/\sqrt{r}$, whereas close to the center, $\varrho(r)\sim const$ and thus $v(r)\propto r$. Good. Now, to get the observed flat rotation curve $v(r)=const$ requires $M(r)\propto r$. Such a mass distribution is what you get for a isothermal sphere, often used as the simplest example of a star in ASTR101 courses. A more detailed analysis and in particular a plethora of studies on intricate n-body simulations favors a modification to that profile called the Navarro–Frenk–White profile.
The take home message is that the dark matter velocity distribution is expected to roughly follow a thermal profile, i.e. a Maxwell-Boltzmann distribution. Modifications come from cropping that at the escape velocity, and from a decade-long debate whether the cores of galactic dark matter profiles are "cored" or "cusped". So, yes, our standard models of dark matter phase space distributions correctly reproduce the observed rotation curves of galaxies. Research is ongoing to investigate feedback mechanisms between the baryonic disk and the dark matter halo, and their impact on the observed distributions and scaling relations.
I can provide some pointers if you want, but to get an idea, have a look at the homepage of the Illustris collaboration. Every galaxy you see there is actually a simulated one, so we know the simulated dark matter profile and can compare this virtual universe to the real one to further our understanding of what is going on, despite not yet having detected dark matter quanta directly. The page also lists a number of papers with details for the so inclined. Or, if you prefer plots, from this first to find though somewhat dated paper comes this example of the velocity distribution in a simulated (dark matter-only) galaxy:

(Maxwell is dot-dashed, dashed and dotted other analytical models, solid black is the simulated profile. Green is what this particular paper propagates as a model, purple an idea about the spread seen in simulations. Also note the inset: as promised, the isothermal halo is a pretty good approximation.
Finally,

Have models been investigated for different motion e.g. where the dark
matter orbits at constant radius, or is stationary, or is even moving
at constant speed towards the galactic nucleus, to be periodically
ejected?

As mentioned dark matter can not be stationary in the galaxy's gravitational potential. The dark matter halo is not expected to condense into a disk, because (a) that would require efficient mechanism for dark matter to dissipate energy and (b) disk-only galaxies aren't stable as already some of the earliest n-body simulations confirmed. Yes, in the thermal halo, the individual dark matter quanta are expected to move on elliptical orbits. Ejection (of notable quantities) would mean evaporation of the dark matter halo which is in contrast to the observation that galaxies are around all the time (i.e. are stable)
