It is space that expands, and it carries matter along with it. In addition to being carried along in space as space expands, matter may also be moving with respect to the space that it in. But what is described as expansion of the Universe is that of space itself by virtue of the fact that it possesses energy over and above that possessed by the matter contained in that space. This energy is described by Einstein's cosmological constant.

For example, the Andromeda galaxy is being carried away from us by the universal expansion of space, but it also has its own local movement through space. In the case of the Andromeda galaxy, it is so close to us that the recession that the expansion of space imparts to it is so small that it is more than offset by its own local motion through space, which happers to be toward us instead of away from us. For much more distant objects, their local movement is overwhelmed by the expansion of space itself, so all of them display red shifts denoting recession.

In addition to the general constant expansion of space, it has recently been determined that space possesses what has come to be called "dark energy" that is causing the expansion of space itelf to accelerate.

As this acceleration causes the expansion rate to increase, it is expected to also affect matter on smaller and smaller scales. Currently, it affects large bodies such as galaxies uniformly, not affecting their contents differentially. But it believed that as time goes on, smaller structures will be affected differentially, so that planetary systems will be disrupted, planets being drawn away from their stars. Then planets will lose their satellites. Then planets will fall apart, then atoms will lose their electrons, then quarks comprising nuclei will separate, leaving only a "quark soup" like what existed back around 10 exp -43 seconds after the Big Bang. We won't be around to witness that.