expansion - Looking for an equation that describes how star (x,y,z) positions are affected by expanding universe

The position of stars change very slowly for a couple of reasons, but not due to an expanding universe.

Galaxies distant from our own, are all moving away from our galaxy at a rate proportional to their distance. The equation is simple:
$$mathrm{speed}=H_0 times mathrm{distance}$$

and $H_0$ is a constant of proportionality, with a value of about 70 km/s per kiloparsec. Since the galaxies are moving away, they don't change their position in the sky at all.

(The nearest galaxies are exceptions to this rule, they may be moving towards us)

Stars in our galaxy are also moving. They have their own proper motion, but the motions of the stars relative to our own are essentially random. You can look up the proper motion of stars on the SIMBAD database.

Stars also appear to wobble due to the annual rotation of the Earth about the Sun and our consequentially changing perspective. The amount of wobble is in inverse proportion to their distance (the nearest stars appear to wobble most)