Not things that are closer than that distance.

The point was, a vast majority of the dark matter in the Milk Way is generated near the center of the galaxy, which is thousands of lightyears away.

Since antimatter and matter will immediately annihilate with any contact, it's a pretty wild thought that a particle could travel that distance, and not come in contact with anything. Even "empty space" is filled with particles. In our solar system, there's about a thousand atoms/cubic meter of "empty" space. Some areas in the galaxy will be less dense than this, while others much more dense (especially when you approach the center of the Galaxy).