Are galaxies moving in each direction with same speed?

Question

All galaxies are moving away from each other (us) because of the expansion of the universe.

Are they moving with same speed in each direction?

Answer 1

Are the galaxies moving towards each direction with same speed ?

Short version : Locally, No. On the cosmic scale, yes-ish.

The motions of galaxies are more complex than this questions suggests.

Let's start with the Milky Way. From our point of view stationary.

But the Milky Way is part of a group of galaxies (the local group). They essentially move in their mutual gravitational fields. Even that field is complicated by the gravitational field of dark matter halos, which are invisible (unobservable directly).

But the local group is not the only group out there. The local group is part of an even larger grouping known as the Virgo Supercluster. Again motions of these groups are complex and again dark matter plays a part.

The local group contains 54 galaxies, and the super cluster contains something on the order of 100 such groups. That's about 5400 galaxies in a fairly complex motion. There's no one easy direction to spot here.

But the larger scale universe is thought governed by what is known as the FLRW metric, which (with dark matter, dark energy and some adjustments) fits the overall expansion of the universe we see.

It's only on that scale (essentially the entire universe excluding the nearest bits) that you see the kind of pattern you mean. So the local bits are maybe 5400 galaxies, but the rest of the universe is very roughly two trillion galaxies. So our local part is on our scale vast and unimaginable, but on the scale that FLRW applies it's a microscopic dot lost in the averages.

FLRW is a result of using General Relativity to model the expansion of the universe. It does predict (when you match it to our universe's conditions) that from any point everything else will be moving away from you (and we now think accelerating away).

There is an apparent contradiction between these two extreme (local group and universe away from that). The resolution of this is that FLRW ignores local grouping and irregularities and assumes a nice continuous distribution (galaxies don't even feature on this grand scale). On that local scale gravitational fields (currently) are large enough to overcome the expansion, hence our closest galaxy (Andromeda) is approaching us, not moving away. But the further away you get, the larger the overall effects predicted by FLRW become and dominate "normal" gravitational effects.

FLRW does not say that they all move away at the same speed. It's distance dependent - further away = faster moving. So not the same speed, but from every viewpoint in the universe the same pattern of faster when further away.