The increasing popularity of Software as a Service has strongly affected the established business model of on‐premise enterprise software. Software as a Service has distinctive characteristics of disruptive innovations that typically create several difficulties for incumbent firms, in particular with regard to adapting business models. To date, however, little empirical understanding exists regarding the dynamics of business model change – a topic of special importance to the highly dynamic software industry, which is characterised by rapid and regular emergence of disruptive innovations. As disruptive innovations require gathering distant knowledge and experimenting with new ideas, this study addresses theoretical gaps regarding the role of absorptive capacity and organisational ambidexterity in the process of business model change owing to the emergence of a disruptive innovation. Drawing on evidence from multiple case studies of six incumbent vendors of enterprise resource‐planning software and informed by a thorough review of related secondary data, we investigate the pace and path of incumbents' business model adaptations. We propose a theoretical model that refines prior literature on absorptive capacity and organisational ambidexterity, particularly with regard to the process of business model change. This study identifies further technological factors that determine how and why incumbents change business models. In addition, our study provides in‐depth insights on the technological trajectory of enterprise resource‐planning software switching from on‐premise to on‐demand software services. © 2016 Blackwell Publishing Ltd