In a gravity assist ( to be technical using a hyperbolic orbit of a large mass to change direction and gain velocity) the object is still following the curvature of space time. So the change in direction is affecting all particles in the object at the same time.
It is the same as people in the ISS orbiting the earth, they do not have to lean in the curve as they are following the curvature in spacetime around the planet. The only difference is one is a hyperbolic orbit (gravity assist) while the other is a parabolic elliptical orbit (ISS)
bjoern_tantau