Yeah, it does make sense, it's just a regular classification task at that point. You should have enough data assigned to clusters though before moving on to neural network.
On the other hand, why don't you predict clusters for new points instead of updating them (you can see separate methods for fit and predict in sklearn's docs, though it depends on technology you are using)? Remember, that neural network will only be as good as it's input (K-Means clusters) and it's predictions will probably be similiar to K-Means.
Furthermore, NNs are more complicated and harder to train, maybe those shouldn't be you first choice.
You could check the idea of fuzzy clustering as well, as the data is fluid it might be a better fit for your case. Maybe autoencoders, as a method of obtaining latent variables, might be of use as well.