Flyway Desktop has native Git integration to push/pull/commit/branch directly from Flyway Desktop.  This is what most of our users have adopted or are moving towards (and most developers in general have adopted). 

If you haven't made the switch yet or you're not planning to, that's ok.  You can stay with the version control systems that you are using, including:

• Team Foundation Server (TFS) - Team Foundation Version Control (TFVC)
• Subversion (SVN)
• Mercurial
• Vault
• Anything else

Use your own version control system's command line or GUI tools (e.g., Visual Studio for TFS, Tortoise for SVN, etc.) to get a local checkout.  You can then point Flyway Desktop to this checked-out directory.  When you "save changes" in Flyway Desktop, it will be written to this directory.  You can then check-in your changes using your own version control system's tools.