Rising and falling doesn't just correlate to social climbing; it also refers to the age old concept of the Wheel of Fortune (the good old Rota Fortunae, for you Latin-lovers), wherein your life gets better as you're raised to the top of the wheel and progressively crappier as you get closer to the bottom.
When the Duchess is finally executed, the main kneeling-standing cycle of the play is completed. In her final moments, without any prompting from her executioners, the Duchess kneels down and welcomes her own death.
It's an intense and complicated moment, and has provoked a wide range of interpretations—what do you think of it? Do you feel like it's a moment of great dignity, because she's accepting the inevitable with calm and grace? Or do you think it's sad; that she's being reduced to the submissive, defeated object her brothers want her to be? Your call, Shmoopers.