Maximizing monotone submodular functions under cardinality constraints is a classic algorithmic problem with several applications in data mining and machine learning. In this paper, we study this problem in a dynamic setting with consistency constraints. In this setting, elements arrive in a streaming fashion, and one is interested in maintaining a constant approximation to the optimal solution while having a stable solution (i.e., the number of changes between two consecutive solutions is bounded).We provide algorithms in this setting with different trade-offs between consistency and approximation quality. We also complement our theoretical results with an experimental analysis showing the effectiveness of our algorithms in real-world instances.