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Recently in the field of unsupervised representation learning, strong identifiability results for disentanglement of causally-related latent variables have been established by exploiting certain side information, such as class labels, in addition to independence. However, most existing work is constrained by functional form assumptions such as independent sources or further with linear transitions, and distribution assumptions such as stationary, exponential family distribution. It is unknown whether the underlying latent variables and their causal relations are identifiable if they have arbitrary, nonparametric causal influences in between. In this work, we establish the identifiability theories of nonparametric latent causal processes from their nonlinear mixtures under fixed temporal causal influences and analyze how distribution changes can further benefit the disentanglement. We propose \textbf{\texttt{TDRL}}, a principled framework to recover time-delayed latent causal variables and identify their relations from measured sequential data under stationary environments and under different distribution shifts. Specifically, the framework can factorize unknown distribution shifts into transition distribution changes under fixed and time-varying latent causal relations, and under observation changes in observation. Through experiments, we show that time-delayed latent causal influences are reliably identified and that our approach considerably outperforms existing baselines that do not correctly exploit this modular representation of changes. Our code is available at: \url{https://github.com/weirayao/tdrl}.