As Marko said, it depends on the algorithm and should be done in the same way the clusters are assigned there ...if you are using R: some clustering packages like e.g. https://cran.r-project.org/web/packages/clustMixType/index.html or the aforementioned ones do provide corresponding predict() methods.
Another question would be whether the new data would lead to similar clusters (i.e. the comparison of two clusterings).
It depends on the algorithm and the dataset to be used. For a dynamic implementation, the data can be considered as a single cluster and based on the training, the clusters can be modified.
You might be interested in an out-of-sample extension. This is to cluster points that have not been part of the initial clustering process. This extension is helpful when not all data is available at the time of the initial clustering, or in applications where the data-set is too large. I attached a couple of interesting links.
Article Kernel Spectral Clustering: Model Representations, Sparsity ...
Article An Out-of-sample Extension of Sparse Subspace Clustering and...