If an astronaut moved away from the earth towards the moon, how would the gravitational forces between the earth and the astronaut change? How would the gravitational forces between the moon and the astronaut change?

As the astronaut moved away from the Earth, the astronaut would experience less gravitational force from the Earth, and the Earth would experience less of a pull from the astronaut (what little pull the astronaut actually produces.) As the astronaut approached the moon, the gravitational pull between them would increase.