Software Adaptation aims at composing incompatible black-box components or services (peers) whose individual functionality is as required for the new system. Adaptation techniques aim at automatically generating new components called adapters. An adapter works as an orchestrator and makes the involved peers work correctly together by receiving all messages exchanged in the system and by correcting mis-match between them. A challenging issue in this area is to consider that peers are described with (possibly cyclic) behavioural models and interact asynchronously, that is, exchanging messages via message buffers. The synthesis of adapters in this context is difficult because the composition of peers may result in infinite systems. In this paper, we propose new adaptation techniques, which rely on a property of communicating systems called stability. Stability aims at verifying whether a communicating system exhibits the same observational behaviour from a certain buffer bound on. We also provide adapter generation techniques using process algebra encodings and enumerative analysis techniques.