In general, how do galaxies move?
Two ways at least - rotation of the galaxy and outward expansion
The Milky Way galaxy, at least, seems to rotate about some central point much like the planets in our solar system revolve around the sun. It takes about 200 million years to complete one rotation. Some scientists have hypothesized that this rotation may be linked to mass species extinction events, but this has never been proven.
The other way galaxies move is their outward expansion that began shortly after the Big Bang, 13.7 billion years ago. Galaxies are spreading away from each other as space and time continue to expand.