This paper introduces HOpiPn, the higher-order pi-calculus with passivation and name creation, and develops an equivalence theory for this calculus. Passivation [Schmitt and Stefani] is a language construct that elegantly models higher-order distributed behaviours like failure, migration, or duplication (e.g. when a running process or virtual machine is copied), and name creation consists in generating a fresh name instead of hiding one. Combined with higher-order distribution, name creation leads to different semantics from name hiding, and is closer to implementations of distributed systems. We define for this new calculus a theory of sound and complete environmental bisimulation to prove reduction-closed barbed equivalence and (a reasonable form of) congruence. We furthermore define environmental simulations to prove behavioural approximation, and use these theories to show non-trivial examples of equivalence or approximation. Those examples could not be proven with previous theories, which were either unsound or incomplete under the presence of process duplication and name restriction, or else required universal quantification over general contexts.