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Others have mentioned, everything is moving away from each other, like the surface of an expanding balloon. But that is what we observe, and there is more beyond the observable universe.
There is an idea that our universe might actually be in a black hole in a higher universe. To expand on the balloon simile (pun intended), this would be like a balloon expanding uniformly except in a spot, a bubble appears and expands faster. A bubble within a bubble. Kind of a tumor, an outgrowth universe. Hope I've been illustrative enough.
It's not exactly what you asked, but a higher level black hole is kind of something pulling all matter in our universe instead of pushing.