When an astronaut is inside of a ship accelerating (from engine burns), or decelerating (due to reentry) they experience a tug in a relative direction.
Suppose an astronaut is in a space ship that is about to undergo a very close approach to a large mass body, as part of a planned gravity assist maneuver.
What would the astronaut experience inside the ship? Would the same tugging effect happen during a velocity change from a gravity assist?
If the only acceleration is due to the large mass's gravity, and the mass is not exceptionally large, or exceptionally close (i.e. close approach to a black hole or a neutron star), the astronaut will not experience any noticeable acceleration relative to the spacecraft. Gravity affects the spacecraft and the astronaut nearly identically, and the acceleration of one matches the acceleration of the other. This is identical to the situation for a spacecraft in a closed orbit around a planet, which likewise is continuously accelerating towards the planet's center.
The effect of gravity decreases with distance from a large mass, so there's always a gravitational gradient across a spacecraft. If the gravity gradient is steep enough, or the spacecraft very roomy, an astronaut who moves away from the spacecraft's center of mass will find themselves on a diverging orbital trajectory, which will tend to push them further still from the center of mass. This is called tidal force. The effect is too small to be noticeable in any reasonable case: a dangerously close Jupiter flyby, say 10km above the cloud tops, would experience a gravitational gradient less than a ten-millionth of one g-force per meter of difference in altitude. Even in a kilometer-long spacecraft a human wouldn't notice a tidal force from one end to the other.
Because the force of gravity decreases with the square of distance from the center of mass, the gravitational gradient gets steeper as you get closer to the mass; for exotic large masses like neutron stars or black holes, it's possible to get much closer to the mass. In theory you can get close enough for tidal effects to be human-perceptible; in practice you probably don't want to get that close. Spoiler for a well known 1966 science fiction story:
Larry Niven's 1966 story, "Neutron Star" considers the effect of gravitational tidal force across a long, skinny, indestructible spacecraft making a close approach to a neutron star, in a setting where starship pilots have somehow forgotten about the existence of tidal force.
Rocket thrust and atmospheric drag, unlike gravity, act on the structure of the spacecraft directly, altering its trajectory relative to that of the astronaut, causing the astronaut to get squished toward one side of the spacecraft, hopefully the side with padding. Note that, due to the Oberth effect, a close approach to a planet may also be a good time to use rocket thrust to effect trajectory changes, so active thrust may be combined with gravity assist.
The astronaut will be accelerated by the gravitational pull of the body they are passing, but they won't "feel" it in a qualitative sense like they do during an engine burn. This is because the gravity assist applies a force to the entire spaceship and everything inside in a uniform manner - gravity pulls on every part of you, from your head to your toes. An engine burn, in contrast, only directly accelerates the ship itself, which then transmits a force to the astronaut via the seat they are strapped into. An astronaut can conduct a gravity assist while floating around inside their spaceship and never touch the walls, but an astronaut cannot remain floating inside a spaceship that's firing its thrusters.
This is a result of the fact that humans "feel" proper acceleration rather than coordinate acceleration, which is relative to the local gravitational field. If you and every point of reference you can see is accelerating the same way under the force of gravity, it won't seem like anything is accelerating at all. This is why astronauts on the ISS "feel" like they're in zero gravity, despite the fact that gravity is still 90% as strong as it is on the surface of the planet. Performing a gravity assist/flyby will be exactly the same - you and the spaceship are simply in freefall the entire time, just like the ISS is already.
This assumes that the local gravitational field is, in fact, reasonably uniform over the scale of the ship, and accelerates the ship and the astronaut the exact same way. This is usually a very good approximation, but it can start to break down with a very close flyby or a very massive body. A close flyby of a black hole, for example, could result in significant tidal forces, which would stretch out the astronaut as their feet got pulled harder than their head.
