I recently contemplated about a thing: on a galaxy scale, the speed of gravity (or gravity change) should have a somewhat noticeable effect. There's a slight pull towards a star on the other side which is not there anymore for 50000 years! Of course there's some other star in that place, but I wonder how minuscule that effect is. Would it speed up or slow down rotation of outer regions of galaxy?

Gravity works the same as light - if you see it somewhere, it pulls you in that direction. The fact that it’s somewhere else now is irrelevant, because the now is tightly coupled to the where thanks to relativity.