We derive the general form of the effective equations governing black hole dynamics in the limit of a large number of dimensions D. These split into a universal soap-bubble embedding condition for stationary configurations and a set of nonlinear dynamical evolution equations describing near-horizon fluctuations of O(1/D) amplitude over horizon scales of \( O\left(1/\sqrt{D}\right) \) . We obtain these equations in full generality, including arbitrary asymptotic sources in the near-horizon region, and we show that they form a parabolic system with a well-posed initial value problem. To connect the various approaches to large-D black hole dynamics, we also show that both the embedding and dynamical equations can be derived from the covariant membrane formalism. We clarify the intrinsic scope of the large-D approach, emphasizing that it yields a well-posed dynamical evolution only on horizon scales of \( O\left(1/\sqrt{D}\right) \) , which is the range where the most relevant horizon dynamics occur. Our results highlight the versatility of these effective theories for studying a wide class of black hole phenomena.
