We present sound and complete environmental bisimilarities for a variant of Dybvig et al.'s calculus of multi-prompted delimited-control operators with dynamic prompt generation. The reasoning principles that we obtain generalize and advance the existing techniques for establishing program equivalence in calculi with single-prompted delimited control. The basic theory that we develop is presented using Madiot et al.'s framework that allows for smooth integration and composition of up-to techniques facilitating bisimulation proofs. We also generalize the framework in order to express environmental bisimulations that support equivalence proofs of evaluation contexts representing continuations. This change leads to a novel and powerful up-to technique enhancing bisimulation proofs in the presence of control operators.