This paper develops the third stage of the Structural Admissibility Regimes (SAR) framework by introducing capacity flow and regime transitions within the finite-capacity admissibility structure established in SAR I and the ordered refinement framework developed in SAR II. We analyze how admissible refinements reorganize the compatibility structure permitted under finite structural capacity. Ordered sequences of non-commuting and non-invertible refinements generate monotonic reduction of admissible commuting closure, producing intrinsically directional refinement flows. This establishes a structural form of irreversibility that arises purely from finite admissibility capacity and ordered refinement composition. Within this framework, regime transitions occur when admissible ordering or compatibility structure can no longer be maintained due to capacity saturation or ordering incompatibility. Geometry, temporal order, and irreversible structural flows therefore emerge as distinct regimes within a unified admissibility framework. Part of the Structural Admissibility Regimes (SAR) foundational series, building on Structural Admissibility (SAR I) and Structural Admissibility (SAR II).