There may not be any center, just as there is no center to the surface of a sphere (there is a center to the volume of a sphere, but not its surface). Either the universe is infinitely large, or it is finite, but neither implies there is a center to it.

There is no way to measure how distant galaxies are moving ACROSS our line of sight. It is possible (through Doppler Effect) to determine how fast they are moving IN our line of sight, towards or away from us. Doppler is extremely precise, very small velocities can be measured accurately, and if we assume tangential velocities are comparable to radial ones, then the galaxies are just too far away and moving too slow for any motion to be detectable, even over thousands of years of continuous observation.

The universe is expanding, and the most distant detectable objects are moving away from us at velocities of an appreciable fraction of the speed of light. If distant galaxies also had a rotational component relative to us, even at speeds comparable to the expansion rate, we would have to measure their motion relative to even more distant stationary background objects that did not share the rotation. Even if the outer edges of the cosmos were rotating near the speed of light, and if there was some fixed reference frame to see them moving in front of, we would still have to wait for millions of years for enough of a parallax to reveal the motion.

I think the current guess is that the universe is extremely large, but finite, and that there is no center and no edge, nor any fixed reference frame (e.g., a center). This is very difficult for us to visualize non-mathematically, just as the surface of the earth seems flat to us no matter where on it we happen to be standing, and no matter how far we walk we never reach the edge.

Other than that, it's turtles all the way down.