Migrating code in TFS from one Team Project to another Team Project within the same Team Project Collection

My company has frequently had the issue of moving code around within the same Team Project Collection as shifting desires and motivations for the organization of our code have swayed our company. Previously, we've used tools such as the TFS Integration Platform available on code plex. That project has its share of issues, namely that:
a) It was half baked to begin with
b) It's no longer maintained and doesn't work for versions later than TFS 2012.

Fortunately for me, thanks to a fortuitous Stack Overflow post, I've found the following script to recursively move the content of one folder to another folder, and it even works across Team Projects within the same collection (but not across Team Project Collections):

Param(
    [Parameter(Mandatory = $true)][ValidateNotNullOrEmpty()][string] $SourceFolder,
    [Parameter(Mandatory = $true)][ValidateNotNullOrEmpty()][string] $TargetFolder
)

Get-TfsChildItem $SourceFolder |
    select -Skip 1 |  # skip the root dir
    foreach {
        & "C:\Program Files (x86)\Microsoft Visual Studio 12.0\Common7\IDE\TF.exe" rename $_.serveritem $_.serveritem.replace($SourceFolder, $TargetFolder)
    }

In order to run the script above, you'll need both Visual Studio and TFS Power Tools installed. When you install the Power Tools, you'll need to ensure that the PowerShell module is selected in the installation options. Also, because the path to the Power Tools PowerShell module isn't put in the PSModulesPath by default, you'll have to manually import the module into your PowerShell settings like so (before you run the script):

Import-Module "C:\Program Files (x86)\Microsoft Team Foundation Server 2013 Power Tools\Microsoft.TeamFoundation.PowerTools.PowerShell.dll"

How to move a shelveset from one branch to another in TFS

You're going to have to start by installing the TFS PowerTools for TFS 2013 (or whatever version is appropriate for you). Once you have the powertools installed and the tools added to your PATH (if they're not already added by the installer), run a command similar to the following in the command line:

tfpt unshelve /migrate /source:"$/ProjectName/Branch" /target:"$/ProjectName/Targetbranch" "My Shelveset Name"

This will unshelve your shelveset into your new branch for you.

WARNING: Before you do this, make sure you're unshelving into a clean branch with no Pending Changes, otherwise you could inadvertently overwrite or otherwise lose important changes.