We put forward a new approach to studying issue definition within the context of policy diffusion. Most studies of policy diffusion---which is the process by which policymaking in one government affects policymaking in other governments---have focused on policy adoptions. We shift the focus to an important but neglected aspect of this process: the issue-definition stage. We use structural topic models to estimate how policies are framed and how these frames vary as a function of prior policy adoptions. Focusing on restrictions on smoking in U.S. states, our analysis draws upon an original dataset of more than 3.1 million paragraphs from newspapers covering 49 states between 1996 and 2013. We find that frames are related to prior adoptions within a state's diffusion network, but this holds true only for frames regarding the policy's concrete implications and not those considering normative justifications. These findings open the way for a new perspective to studying policy diffusion in many different areas.