You can use TFS Integration Tools to move team project in TFS2013, but there are also some limitations. Detail for you reference: https://visualstudiogallery.msdn.microsoft.com/eb77e739-c98c-4e36-9ead-fa115b27fefe
I have test in my local environment , succeed.(Both different collection and the same collection)
My environment:
TFS2013 SQL2012 TFS 2010 Team Explorer VS2013 TFS Integration Tools2012
Prepare environment:
Install TFS2010TeamExplorer Cause TFS Integration Tools2012 need it
Install TFS Integration Tools2012
PS: It’s not support SQL2014. When you connect SQL2014 during the install of Integration Tools , it would throw out an error: Vaild values of the database compatibility level are 100.110,or120... SQLkey:CreateDatabaseScript…90
Detailed Steps:
- Create a new empty team project in another collection. Choose the same Process Template as source project when creating new project.
- Launch TFS Integration, Click Create New->Choose "VersionControlAndWorkItemTracking" template under "Team Foundation Server" folder (for you could use 'VersionControl' template)
- Configure Workflow Type, Frequency, Version Control Session as below, Left Source should be the project you want to copy, Right Source should be the empty new project
Note, for the Paths section, you must include any folders (paths)
under the root explicitly and individually in the folders list. Just
using $/ or $/MyProject will not work. For example, if you have
folders src/ and test/ under the root, you must include these as
$/MyProject/src and $/MyProject/test/
- Do the same for Work Item Tracing Session.
- Click Save to Database button. Then click Start.
- Resolve the conflicts if any.
- Done

At last you can view your history in the Changeset Comment. For example: It’s created of Changeset134 2015-10-19
