The increasing number of digital devices in our environment enriches how we interact with digital content. Yet, cross-device information transfer – which should be a common operation – is surprisingly difficult. One has to know which devices can communicate, what information they contain, and how information can be exchanged. To mitigate this problem, we formulate the gradual engagement design pattern that generalizes prior work in proxemic interactions and informs future system designs. The pattern describes how we can design device interfaces to gradually engage the user by disclosing connectivity and information exchange capabilities as a function of inter-device proximity. These capabilities flow across three stages: (1) awareness of device presence/connectivity, (2) reveal of exchangeable content, and (3) interaction methods for transferring content between devices tuned to particular distances and device capabilities. We illustrate how we can apply this pattern to design, and show how existing and novel interaction techniques for cross-device transfers can be integrated to flow across its various stages. We explore how techniques differ between personal and semi-public devices, and how the pattern supports interaction of multiple users.