Understanding the diffusion of ideas and behaviors is essential in the social and biomedical sciences. Despite its importance, estimating causal diffusion effects, also known as peer or contagion effects, is difficult due to contextual confounding and homophily bias. To address this long-standing problem, we examine the identification of causal diffusion effects under a new assumption of structural stationarity, which formalizes the underlying diffusion process with a class of dynamic causal directed acyclic graphs. First, we develop a statistical test that can detect a wide range of biases, including the two types mentioned above. We then propose a difference-in-differences style estimator that can directly correct biases under an additional parametric assumption. Applying the methods to fine-grained geo-coded hate crime data, we find that the diffusion of hate crimes is concentrated in areas with a high proportion of school dropouts, which has practical policy implications.