This article provides information on how you will be able to continue pushing changes to your repository when Kiln gets stuck and shows this message:
Kiln stuck on 'Incoming push! Please refresh in a moment. Processing changesets.'
While there are several reasons why the processing changesets message can get you stuck while working on a repository, most of these reasons are self-healing.
If you encounter this message, find a recent push to that failed and re-attempt to push the changes until you get a successful one (or any subsequent changes).
Alternatively, you can also pull the repository again cleanly (clone) on another directory or machine, which should have all the files from the last successful push. After that, you can continue working on this new location.