How can I migrate all source code from one Visual Studio Team Services project to another?
You could use Tfs Integration tools to achieve it.
Detail steps: Move Team Project to another Project Collection TFS 2013
Note: Its a lossy data transfer.
Migration - For this project, and for TFS in general, we use the term "migration" to refer to moving data between two different servers. Migration is done through the APIs of TFS, and is a lossy data transfer (see the limitations listed below).
Source & Limitations list:https://visualstudiogallery.msdn.microsoft.com/eb77e739-c98c-4e36-9ead-fa115b27fefe
If you just need to migrate the code, then simply Git clone and push to another remote. See here to do that with Git command line: https://stackoverflow.com/a/10312587/383029