Switch to a dropped stage

Fly a piece you just dropped instead of the ship you launched with: a spent booster with a Sat Controller aboard, or any other piece that can steer itself.

You need this to fly a booster home after it separates, or to check on a piece that carries its own command part before it falls out of reach.

A stage carrying a Command Pod, a Crew Capsule or a Sat Controller has already separated. A piece with none of the three is plain debris and cannot be flown.
You are in the flight, not on the map or in the hangar.

Steps

Look for the ⇄ button near the turn controls at the bottom of the flight screen. It only appears once a piece nearby carries a command part of its own.
The ⇄ button appears the moment a flyable piece is nearby.
Tap it, or press V on a keyboard. Control jumps to that piece: its own engines, fuel and staging take over the screen, and the ship you were just flying is left to coast on whatever it was doing.
Now flying the booster: its own stage, its own throttle.
Tap ⇄ again to come back. It returns you to the ship you just left, exactly as you left it.

What you should see

The fuel, staging and throttle readouts all belong to whichever ship you are flying right now. Whatever you left behind keeps going on its own: a booster still burning keeps burning until its tank runs dry, an open parachute keeps dragging, and none of it can be steered until you switch back to it. A piece is solid, so a burning one that drifts back into your ship can still strike it.