EDIT: The teb_local_planner package has now support for dynamic obstacles based on a constant velocity model.
You can find two new tutorials on that topic in the tutorials section.
The navigation stack currently does not support dynamic obstacles actually. However, dynamic obstacles are treated as static in the current control/sampling interval. Due to the discretized grid representation of the costmap any temporal coherence isn't capured. However, extending the DWA or teb_local_planner to dynamic obstacles is in principal straight forward, but it would require additional tracking and motion prediction on top of the costmap.
Up to my knowledge, there isn't any local planner for the navigation stack that performs it's own tracking and prediction.
I am not aware of other planning packages beside of the navigation stack.
However, you can improve the current obstacle avoidance behavior by adding the inflation layer to the local costmap (in addition to the global costmap). By doing so, the robot starts earlier avoiding the obstalce (w.r.t. the static snapshot).
Currently, the teb_local_planner does not take inflated cells into account. In that case, refer to the DWA planner.
A concept similar to the original inflation has already been implemented in the teb_local_planner package
There are multiple other questions here with similar concerns: