Is it true that many galaxies are starting to slowly move to one point in space

An what is the Galactic Core, I know it's probably a Black Hole, but could it lead into an alternate Universe?

You're curious, good. The first guy made a point, but he didn't emphasize it. Galaxies are flying away from each other as long as the gravity between them is weak. Imagine if space could grow. (hint it is..kinda) Then imagine that it is growing. That would mean that all the galaxies are floating around and the space between them is growing. This is a linear function and because of that galaxies which are billions of light years away are flying away MUCH faster than galaxies closer to us. Look up Hubble Law, fascinating stuff. Also, never ever tell anyone that the Big Bang was some kind of explosion, the truth is far more amazing than that.

Which galaxies? I don't think a black hole large enough to influence galaxies could be alone in space. Large black holes are part of galaxies because that is where the mass for the black hole comes from. Did you know that the Milky Way is currently colliding with another galaxy. It is called Canis Major Dwarf.

There is a region of space that is called the Great Attractor, because it seem to have a mass of several thousand times that of our galaxy and would pull most galaxies around it towards itself.
But
1- it has not just started to do so, it has been at it for billion of years
2- it is not the universe center, or a galactic core
3- it may or not be a black hole, as this does not change anything. A black hole with the mass of a galaxy would act and exhibit a gravitational pull on neighboring bodies that is the exact same as a galaxy of the same mass. A black hole is simply more compact.

As for the leading to an alternate universe, there are mathematical simulation that show the inside of a black hole would create complete universes as complex and large as our own, with its own timeline, but in dimensions that our perpendicular to ours, so they would forever be impossible to observe.