Mutations in non-coding regulatory DNA such as enhancers underlie a wide variety of diseases including developmental disorders and cancer. As enhancers rapidly evolve, understanding their function and configuration in non-human disease models can have important clinical applications. Here, we analyze enhancer configurations in tissues isolated from the common marmoset, a widely used primate model for human disease. Integrating these data with human and mouse data, we find that enhancers containing trait-associated variants are preferentially conserved. In contrast, most human-specific enhancers are highly variable between individuals, with a subset failing to contact promoters. These are located further away from genes and more often reside in inactive B-compartments. Our data show that enhancers typically emerge as instable elements with minimal biological impact prior to their integration in a transcriptional program. Furthermore, our data provide insight into which trait variations in enhancers can be faithfully modeled using the common marmoset.