What is causing asteroids to change their paths when they come close to Earth?

There are several things in space that knock asteroids around. One is dust. There might be one molecule in a cubic meter, and anything moving around runs into them. Some of that dust has an electric charge, and the charge varies according to distance from the sun. So an asteroid coming in can be pushed or pulled, depending on the differences in charge. Electric charge is actually a more formidable force than gravity, until two bodies come relatively close together. This link examines electrical effects in space:
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