Molecularly asymmetric triblock copolymers progressively grown from a parent diblock copolymer can be used to elucidate the phase and property transformation from diblock to network-forming triblock copolymer. In this study, we use several theoretical formalisms and simulation methods to examine the molecular-level characteristics accompanying this transformation, and show that reported macroscopic-level transitions correspond to the onset of an equilibrium network. Midblock conformational fractions and copolymer morphologies are provided as functions of copolymer composition and temperature.