We have explained, and shown by feature stringy examples, why a D-brane in superstring theory, when treated as a fundamental dynamical object, can be described by a map from an Azumaya/matrix manifold with a fundamental module with a connection to the target spacetime. In this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional for such pairs when the target spacetime is equipped with a background (dilaton, metric, B)-field from closed strings. We next develop a technical tool needed to study variations of this action and apply it to derive the first variation of with respect. The equations of motion that govern the dynamics of D-branes then follow. We introduce a new action standard for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This ‘standard action’ is abstractly a non-Abelian gauged sigma model based on maps from an Azumaya/matrix manifold with a fundamental module with a connection enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. A complete action for a D-brane world-volume must include also the Chern-Simons/Wess-Zumino term that governs how the D-brane world-volume couples with the Ramond-Ramond fields . The current notes lay down a foundation toward the dynamics of D-branes along the line of this research project.