Classical matrix completion methods focus on data with stationary latent structure and hence are not effective in missing value imputation when the latent structure changes with time. This paper proposes a dynamic nonlinear matrix completion (D-NLMC) method, which is able to recover the missing values of streaming data when the low-dimensional nonlinear latent structure of the data changes with time. The paper provides an efficient approach to updating the nonlinear model dynamically. D-NLMC incorporates the information of new data and remove the information of earlier data recursively. The paper shows that the missing data can be estimated if the change of latent structure is slow enough. Different from existing online or adaptive low-rank matrix completion methods, D-NLMC does not require the local low-rank assumption and is able to adaptively recover high-rank matrices with low-dimensional latent structures. Note that existing high-rank matrix completion methods have high-computational costs and are not applicable to streaming data with varying latent structures, which fortunately can be handled by D-NLMC efficiently and accurately. Numerical results show that D-NLMC outperforms the baselines in real applications.