Latent world models allow agents to reason about complex environments with high-dimensional observations. However, adapting to new environments and effectively leveraging previous knowledge remain significant challenges. We present variational causal dynamics (VCD), a structured world model that exploits the invariance of causal mechanisms across environments to achieve fast and modular adaptation. VCD identifies reusable components across different environments by combining causal discovery and variational inference to learn a latent representation and transition model jointly in an unsupervised manner. In evaluations on simulated environments with image observations, we show that VCD is able to successfully identify causal variables. Moreover, given a small number of observations in a previously unseen, intervened environment, VCD is able to identify the sparse changes in the dynamics and to adapt efficiently. In doing so, VCD significantly extends the capabilities of the current state-of-the-art in latent world models.