The calculation of 3-D motion requires precise position measurements at recorded times.
The change in position is a distance.
The change in position from start to finish defines the 3-D velocity vector.
The change in time from start to finish defines the motion's elapsed time.
The distance divided by the elapse time results in the object's velocity, which has the observed 3-D vector.
Johannes Kepler developed his Laws of Planetary Motion using the precise celestial measurements recorded by Tycho Brahe. Kepler defined the motion of an ellipse which is 3-D motion. The ellipse could be at an angle to the observer.
Planets are millions of km or miles away but their motions can be measured by noting changes in position over a number of years. Using Earth's orbit as a base line, distance measurements are done by parallax.
This process is more difficult beyond the solar system.
Once a star's parallax is known, its distance from Earth can be computed trigonometrically. But the more distant an object is, the smaller its parallax. Even with 21st-century techniques in astrometry, the limits of accurate measurement make distances farther away than about 100 parsecs (or roughly 300 light years) too approximate to be useful when obtained by this technique. This limits the applicability of parallax as a measurement of distance to objects that are relatively close on a galactic scale. Other techniques, such as standard candles and spectral red-shift, are required to measure the distance of more remote objects.
Unfortunately, that spectral red-shift method is unreliable.
In 1936 Edwin Hubble concluded, from the galaxy red shifts being measured, the Local Group is on an island separate from the Hubble Flow. A 1999 paper claimed its radius was confirmed having a radius of 1.18 Mpc. The galaxies beyond our Local Group were in the Inter Galactic Medium where a hydrogen absorption line increased its red shift by distance because of hydrogen atoms in the IGM (at a low density). This conclusion from decades ago was confirmed by several recent studies of the very high red shift galaxies which concluded these high red shifts in the hydrogen absorption line are caused by the hydrogen atoms in the IGM. The highest galaxy red shift is z > 11. They know these galaxies are not moving at many times the velocity of light. Slipher in the 1920's suggested galaxy red shifts were also a velocity. This was a mistake.
The red shift comes from the IGM ( not verified to be uniform over all scales) and is not a velocity. Using a red shift as a distance metric has an unknown margin of error.
Standard candles are mentioned. They were used to calculate the distance to M31 for the first measurement of a galaxy beyond our Milky Way,
M31 is our closest galaxy, ignoring the Milky Way satellites.
The Cepheids in M31 can be used to calculate a 3-D velocity for M31.
Each Cepheid provides both a position and distance.
Making measurements over many years the star's 3-D motion and velocity can be calculated.
If M31 were moving at z=1 then the Cepheid would change its position 1 ly each year.
The distance to M31, using Cepheids, is 2.54 million ly.
The position change year to year will be difficult to resolve.
At that distance and at whatever velocity, everyone should agree any motion by M31 will require many human life-times to observe and measure.
With increasing distance object positions lose precision.
Until these measurements have been made for a long enough time, there is no evidence for any motion by any distant galaxy.
Cepheids have a limited range.
The universe is huge and we can measure relative positions of many distant galaxies. Their distances are uncertain.
The reality is on the galactic distance scale they are essentially not moving. There is no evidence to the contrary.
That simple statement falsifies the claims of universe expansion and the big bang.
Our Milky Way is huge so stars must be observed and measured for years to measure their motion. The Gaia probe did that exercise for over a billion objects.
We recently learned about the motion of many stars in our Milky Way.
The universe beyond our Milky Way lacks that data.
We must stop believing we have it.
Simulations have been developed using only unidirectional red shifts, while lacking any data for transverse motion.
Astronomers need a little humility. We really don't know of any motion by distant galaxies. They are just as we see them. There is still wonder in that.