Objects that close are both red-shifted and blue-shifted...
...depending on whether our relative motion compared to theirs results in a net increase or decrease in distance. Dunno why I made that so complicated. Are they getting closer or further?
I assure you that astronomical objects do not flee from the center of our galaxy. If they are close to it then they are probably in orbit around it (though, the whole galaxy could be said to orbit our central black hole, I guess.). And if they aren't orbiting it, then they are falling into it or (much harder to do) escaping the galaxy.