For distant objects in a curved spacetime, you can define their velocities by means of parallel transport. Choose a spacetime path between yourself and the object, and drag its 4-velocity vector along that path, while preserving the direction of the vector during each infinitesimal step along the path.

The outcome from this process depends on the path that you choose. This is why velocities of distant objects are said to be ill defined. However, if you receive light from the distant object and study its frequency shift, that tells you the object's velocity based on parallel transport along the spacetime path of the light. Note that this is true whether the frequency shift is kinematic (Doppler), gravitational, "cosmological", or whatever. There is no fundamental difference between these frequency shifts -- they are all the same phenomenon viewed from different perspectives.

In practice, for a distant galaxy, it is often convenient to decompose the redshift into a cosmological contribution and a peculiar contribution. The cosmological contribution tells you the distance of the object, in accordance with Hubble's law. The peculiar contribution tells you the motion of the object with respect to its local environment. For an individual galaxy, you can only do this decomposition in principle if you have an independent distance measurement (e.g., a supernova that happened in the galaxy, from which you can infer distance from brightness). However, for a collection of galaxies, you can do it statistically.

For example, if you naively interpret all redshifts as cosmological, then you would obtain a highly anisotropic galaxy distribution, because the galaxy peculiar velocities are distorting their inferred positions along the line of sight, and not in any other direction. However, we expect the galaxy distribution to be statistically isotropic. Thus, you can decompose the statistics of your naively inferred galaxy distribution into isotropic and anisotropic components, and the latter can be re-interpreted as coming entirely from peculiar motion of the galaxies. This is the basic idea behind using redshift-space distortions as a cosmological probe.

For distant objects in a curved spacetime, you can define their velocities by means of parallel transport. Choose a spacetime path between yourself and the object, and drag its 4-velocity vector along that path, while preserving the direction of the vector during each infinitesimal step along the path.

The outcome from this process depends on the path that you choose. This is why velocities of distant objects are said to be ill defined. However, if you receive light from the distant object and interpret its frequency shift as a relativistic Doppler shift, that tells you the object's velocity based on parallel transport along the spacetime path of the light. Note that this is true whether you regard the frequency shift as kinematic (Doppler), gravitational, "cosmological", or whatever. There is no fundamental difference between these frequency shifts -- they are all the same phenomenon viewed from different perspectives.

In practice, for a distant galaxy, it is often convenient to decompose the redshift into a cosmological contribution and a peculiar contribution. The cosmological contribution tells you the distance of the object, in accordance with Hubble's law. The peculiar contribution tells you the motion of the object with respect to its local environment. For an individual galaxy, you can only do this decomposition if you have an independent distance measurement (e.g., a supernova that happened in the galaxy, from which you can infer distance from brightness). However, for a collection of galaxies, you can do it statistically.

For example, if you naively interpret all redshifts as cosmological, then you would obtain a highly anisotropic galaxy distribution, because the galaxy peculiar velocities are distorting their inferred positions along the line of sight, and not in any other direction. However, we expect the galaxy distribution to be statistically isotropic. Thus, you can decompose the statistics of your naively inferred galaxy distribution into isotropic and anisotropic components, and the latter can be re-interpreted as coming entirely from peculiar motion of the galaxies. This is the basic idea behind using redshift-space distortions as a cosmological probe.