Let's say if a galaxy is moving, shouldn't it be shaped like a funnel? I am saying this because it would take considerable time for the outermost star to know that the center of gravity has shifted. For example, if milky way's radius is 100,000 light years, there is possibly no way for the outermost star in our galaxy to know that the center of gravity has moved before 100,000 years. So taking a circular symmetry, the only thing that comes to my mind is a funnel-like shape.
Let's say if a galaxy is moving, shouldn't it be shaped like a funnel? I am saying this because it would take considerable time for the outermost star to know that the center of gravity has shifted.
Moving with respect to what?
Suppose the galaxy's center of mass is moving at some velocity respect to some external object. You can always find some other external object for which the motion of the galaxy's center of mass is in exactly the opposite direction. Or yet another object for which the galaxy isn't moving at all.
This concept of relativity dates all the way back to Galileo. Galilean relativity is implicit in Newtonian mechanics. Einstein went one step further and made relativity a central concept in his theories of special and general relativity.
One thing that is absolute in Newtonian mechanics and special relativity is acceleration. This concept carries forward to some extent to general relativity. Suppose a galaxy is on a collision course with another galaxy. When the galaxies are far removed from one another, all of the stars in one galaxy experiences more or less the same (very small) gravitational acceleration toward the other. But when the galaxies get close to one another, the stars in one galaxy closest to the other galaxy experience a much greater gravitational acceleration than do the stars in that galaxy that remain well removed from the other galaxy.
These tidal accelerations have the potential to tear galaxies apart, but not into a funnel shape.
You are forgetting about Newton's 1st law. Objects in motion stay in motion unless an outside force acts on it. The stars in the galaxy are already moving relative to the center of mass, and so they will keep up with the center of mass by default. They don't need information about the motion of the center of mass of the galaxy because objects don't need forces to make them move at a constant velocity.
It's kind of like asking why I don't drag behind my car when I drive on the highway. Once me and my car are both going 60 mph, there doesn't even need to be a force keeping me going because of Newton's 1st law.