Charged particles can hit the earth at relativistic speeds. But it seems that all large bodies have fairly low relative speed. Of course, speed can increase considerably when a body orbits close to a massive object, but then it will not travel very far, and it can be discounted by averaging the speed over a long enough time (maybe a year).
The Earth cruises around the Sun at 30 km/s, the sun cruises at 200 km/s and the Milky Way at 600 km/s. Not that much.
I have two somewhat opposite questions:
What do we know of the relative speeds of massive bodies in the universe, small or large (not particles)? I am not sure whether it was the proper way to state the question. Also, do relative speeds get very high if we correct out the part due to universe expansion? Are there braking phenomena?
On the other hand, what initiated that motion? If the initial soup had been homogenous, the coalescence of randomly moving particles should have produced structures essentially at rest (ahem: where is the energy going?) with respect to each other. There were small variations in temperature, but how should that create speed differences? Or did it cause large scale streams that coalesced into moving bodies?
Even if some speed is due to contraction of large scale rotating structures, it does require initial momentum to exist.
To put it together, do we have measured speed statistics in conformance with universe evolution models? What does it say about speed?
Sorry if the questions are not well stated; that is my best. A reference to a paper for non-specialists would do too.
Added after 3 weeks, considering the answer by @BenCrowell and the comment of @ChrisWhite (thank you both).
Please, forget the part about cosmological speed comparisons.
This is my own answer to my question; I leave it as the question, since it raises other issues. The first point is: does my answer make sense? It is only guesswork on my part. I wrote this answer because, though really helpful, Ben Crowell's answer did not really answer the heart of my initial question.
Though I suggested it in my initial question, I realize even more now that the problem is only the origin of momentum, probably only angular momentum. This was confirmed by the answer of Ben Crowell regarding the fact that structures were essentially at rest to start with.
Probably my main error was to think that there might be another "source of speed and momentum", and my awkward attempt to discount observed very high speeds as going nowhere because in tight orbit.
I do not see how large scale momentum could come from some form of accretion of spontaneously emerging angular momentum quanta. I would think they would balance on average without any visible macroscopic effect, outside very special and anisotropic places like a black hole horizon (and even then). I feel the same regarding linear momentum quanta (if such a thing exists).
My guess is that high scale momentum arises from momentum exchange between large collapsing structures. It is well known that celestial structure, such as planets and satellites, can exchange angular momentum. Though I no longer try to follow the mathematical analysis, I also read that much of the exchange can be mediated by tidal effects. But tidal effect should be even stronger when it is between structures that are not yet collapsed and are hence very deformable. Angular Momentum exchange is not necessarily one structure slowing while another is speeding up. It is vectorial and may be two structures both speeding up in opposite directions, provided extra energy gets in from somewhere, such as potential energy from collapse.
So it would be the case that, as they are collapsing, structures are being deformed under the gravitational influence of other structures, so that instead of converging to some global center of gravity, some subparts collapse separately around their own centers of gravity, and rotate around the main center of gravity.
Is it actually in that way that the originally small variations of temperature (density?) created the various structures of the universe?
Observed momentum arising from such momentum exchanges, there is no reason it should have the same orientation in all subparts of a larger structure, and this is indeed observed in the solar system. (see Can the axis of rotation of a celestial body point in any arbitrary direction?) What about other structures in the universe?
The variations in speed (earth in solar system, sun in galaxy, etc.) are obviously just the consequence of variations in mass and size of the structures, very massive structure permitting higher speed as observed. The speed formula being $v = \sqrt{\frac{G\cdot M} r}$, and the mass most likely increasing like the cube of the (initial) structure size.
Hence the "average speed" in a structure should grow more or less like the initial linear size of the structure (assuming an almost uniform initial density). As reported in the initial question above, this is not quite what is observed. Where do I err?
Could it be that very large structures collapse less than smaller ones because of the angular momentum of the subpart? This could reduce the influence of the total mass on individual subparts, and increase the orbital radius, thus reducing the speed.
Is there any way to measure the degree of collapse as a function of initial size of a structure, and some notion of reference speed of substructures?
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