In my mind, I picture a gravity assist sling shot boost like a ball on the end of string being swung until the centrifugal force grows strong and then is released. Where gravity is the string, the planet is the swinger, and ship is the ball.
If your ship is under acceleration when performing a gravity assist, would there be centrifugal force applied? Or asked another way, what would be the perceived gravitational increase to the occupants/cargo of ship using gravity slingshot while under thrust?