There are five space treaties: https://www.unoosa.org/oosa/en/ourwork/spacelaw/treaties.html. Article II of the Outer Space Treaty states no one owns moons or other celestial bodies, asteroids including. The Moon Agreement, the last of the treaties states it covers more than just the Moon and Article VII, Section I talks about disrupting the environment of the object, but it seems more geared towards changes and/or contamination.

I’m not sure, but maybe the idea of colliding with an asteroid to change its trajectory was not around at the time the space treaties were drafted. So outside of getting the proper approval for launch, registering the spacecraft, etc… there does not appear to be a policy or law governing this.

Definitely something that should be addressed in future space law policies. Think about this: there is the Liability Convention and the Registration Convention space treaties. DART is registered and what happens if fragments from the collision cause damage to another spacecraft, hurts an astronaut, or is big enough to reach Earth’s surface to cause damage?