In real-world applications, data are often collected in the form of a stream, with features that can evolve over time. For instance, in the environmental monitoring task, features can be dynamically vanished or augmented due to the existence of expired old sensors and deployed new sensors. Additionally, besides the evolvable feature space, the labels potentially contain noise. When feature space evolves and data conceal inaccurate labels at the same time, it is quite challenging to design algorithms with guarantees, particularly theoretical understandings of generalization ability. To address this difficulty, we propose a new discrepancy measure for noisy labeled data with evolving feature space, named the label noise robust evolving discrepancy. Using this measure, we present the generalization error analysis, and the theory motivates the design of a learning algorithm which is further implemented by deep neural networks. Empirical studies on synthetic data confirm the rationale of our discrepancy measure and extensive experiments on real-world tasks validate the effectiveness of our algorithm.