The standard Komar mass provides an elegant, quasi-local measure of total gravitating energy for stationary spacetimes, but it conventionally fails in dynamical scenarios that lack an exact global timelike Killing vector field. In this paper, we explore a generalized Komar energy integral bounded by a spacelike two-surface, obtained by replacing the Killing vector with the purely kinematic normal evolution vector field associated with a 3+1 spacelike foliation. Through a step-by-step mathematical derivation, we show that this generalized integral reduces to the spatial boundary flux of the Eulerian 4-acceleration. Furthermore, by evaluating the asymptotic vacuum constraints in generic dynamically settling spacetimes without restricting the metric to an isotropic or transverse-traceless spatial gauge, we establish, at leading asymptotic order, a relationship mapping this flux to the global Arnowitt-Deser-Misner (ADM) mass. By constructing a superpotential for the linearized spatial Einstein tensor, we prove that the dynamically generalized Komar energy and the ADM mass are equivalent up to a boundary flux of the temporal evolution of the conjugate momentum. We verify this relationship through non-trivial analytical test cases involving dynamical gauge foliations and physical gravitational wave radiation, revealing that the geometric momentum flux isolates longitudinal time-slicing artifacts while naturally decoupling from transverse radiative degrees of freedom.