  We show that the holographic Complexity = Volume proposal satisfies a very
general notion of Momentum/Complexity correspondence (PC), based on the
Momentum Constraint of General Relativity. It relates the rate of complexity
variation with an appropriate matter momentum flux through spacelike extremal
surfaces. This formalizes the intuitive idea that `gravitational clumpling' of
matter increases complexity, and the required notion of `infall momentum' is
shown to have a Newtonian avatar which expresses this idea. The proposed form
of the PC correspondence is found to be exact for any solution of Einstein's
equations in 2+1 dimensions, and any spherically symmetric solution in
arbitrary dimensions, generalizing all previous calculations using spherical
thin shells. Gravitational radiation enters through a correction which does not
have a straightforward interpretation as a PC correspondence. Other
obstructions to an exact PC duality have a topological origin and arise in the
presence of wormholes.
