What does general relativity say about the relative velocities of objects that are far away from one another?
Nothing. General relativity doesn't provide a uniquely defined way of measuring the velocity of objects that are far away from one another. For example, there is no well defined value for the velocity of one galaxy relative to another at cosmological distances. You can say it's some big number, but it's equally valid to say that they're both at rest, and the space between them is expanding. Neither verbal description is preferred over the other in GR. Only local velocities are uniquely defined in GR, not global ones.
Confusion on this point is at the root of many other problems in understanding GR:
Question: How can distant galaxies be moving away from us at more than the speed of light?
Answer: They don't have any well-defined velocity relative to us. The relativistic speed limit of c is a local one, not a global one, precisely because velocity isn't globally well defined.
Question: Does the edge of the observable universe occur at the place where the Hubble velocity relative to us equals c, so that the redshift approaches infinity?
Answer: No, because that velocity isn't uniquely defined. For one fairly popular definition of the velocity (based on distances measured by rulers at rest with respect to the Hubble flow), we can actually observe galaxies that are moving away from us at >c, and that always have been moving away from us at >c.[Davis 2004]
Question: A distant galaxy is moving away from us at 99% of the speed of light. That means it has a huge amount of kinetic energy, which is equivalent to a huge amount of mass. Does that mean that its gravitational attraction to our own galaxy is greatly enhanced?
Answer: No, because we could equally well describe it as being at rest relative to us. In addition, general relativity doesn't describe gravity as a force, it describes it as curvature of spacetime.
Question: How do I apply a Lorentz transformation in general relativity?
Answer: General relativity doesn't have global Lorentz transformations, and one way to see that it can't have them is that such a transformation would involve the relative velocities of distant objects. Such velocities are not uniquely defined.
Question: How much of a cosmological redshift is kinematic, and how much is gravitational?
Answer: The amount of kinematic redshift depends on the distant galaxy's velocity relative to us. That velocity isn't uniquely well defined, so you can say that the redshift is 100% kinematic, 100% gravitational, or anything in between.
Let's take a closer look at the final point, about kinematic versus gravitational redshifts. Suppose that a photon is observed after having traveled to earth from a distant galaxy G, and is found to be red-shifted. Alice, who likes expansion, will explain this by saying that while the photon was in flight, the space it occupied expanded, lengthening its wavelength. Betty, who dislikes expansion, wants to interpret it as a kinematic red shift, arising from the motion of galaxy G relative to the Milky Way Galaxy, M. If Alice and Betty's disagreement is to be decided as a matter of absolute truth, then we need some objective method for resolving an observed redshift into two terms, one kinematic and one gravitational. But this is only possible for a stationary spacetime, and cosmological spacetimes are not stationary. As an extreme example, suppose that Betty, in galaxy M, receives a photon without realizing that she lives in a closed universe, and the photon has made a circuit of the cosmos, having been emitted from her own galaxy in the distant past. If she insists on interpreting this as a kinematic red shift, the she must conclude that her galaxy M is moving at some extremely high velocity relative to itself. This is in fact not an impossible interpretation, if we say that M's high velocity is relative to itself in the past. An observer who sets up a frame of reference with its origin fixed at galaxy G will happily confirm that M has been accelerating over the eons. What this demonstrates is that we can split up a cosmological red shift into kinematic and gravitational parts in any way we like, depending on our choice of coordinate system.
For those with a more technical background in abstract math, the following description may be helpful. (The answer by knzhou does a nice job of explaining this in nontechnical terms.) Spacetime in GR is described as a semi-Riemannian space. A velocity vector is a vector in the tangent space at a particular point. Velocity vectors at different points belong to different tangent spaces, so they aren't directly comparable. To compare them, you need to parallel transport them to the same spot. If the spacetime is (approximately) flat, then you can do this, and you can say, for example, that the sun's velocity vector minus Vega's velocity vector is a certain value. But if the spacetime is not even approximately flat (e.g., at cosmological scales), then parallel transport is path-dependent, so the comparison becomes completely ambiguous.
Related: Why is the observable universe so big?
Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf