It depends on how strong the gravity source is and how close you get to it. The strength of gravity falls off with distance following the inverse-square law, so different parts of your spaceship can feel different amounts of force acting on them during the manoeuvre. In the clip, there's a large but weak source of gravity (a moon), so I don't know how much force the pilot's going to feel - probably less than he feels because the ship is rotating and generating centrifugal force.

But we've got a pretty good example of what I'm talking about over our heads - Earth's moon is in orbit around us. You could think of that as a gravitational assist manoeuvre that never ends because the moon didn't gain enough speed to break free of Earth's gravity. Over a time scale of billions of years, the difference between the gravitational force exerted on the near side and the far side is causing the moon's rotation to slow down.

For a fictional example that makes it clearer, Larry Niven wrote a story in 1968 titled "There Is A Tide" in which a pilot has a close encounter with a neutron star.