The case opened when the user tried to install an application during the upgrade task sequence. Unfortunately for him, that did not work and there were no indications that MDT even attempted to initiate application installation. The first thing the user did was to check if the application would install while using a standard "Client Deployment Tasksequence". He quickly realized that the application would install normally and this is where he turned to TechNet for help and I started digging a bit deeper.
I began by taking a look at the standard MDT In-Place upgrade task sequence. After opening the task sequence editor I selected the "Post-Processing" group and started poking around conditions that MDT must evaluate before running task sequence steps assigned to this group. I quickly came upon the registry condition, that ensures that MDT executes this task sequence group only if the value of the registry key Software\Microsoft\Windows NT\CurrentVersion\Windows\Win10UpgradeStatusCode equals SUCCESS.
I suspected that the underlying issue was that the registry value was not being set which in turn meant that MDT would skip processing the task sequence group "Post-Processing". This obviously wasn't a good sign, but it wasn’t the worst of it: usually %DeployRoot%\Scripts\SetupComplete.cmd script is supposed to set this value, however in this case the script was missing the relevant bit:
reg add "HKEY_LOCAL_MACHINE\Software\Microsoft\Windows NT\CurrentVersion\Windows" /v Win10UpgradeStatusCode /t REG_SZ /d "Success" /f
After some discussion with the user I had a pretty solid idea what happend: the user was still running Microsoft Deployment Toolkit 2013 Update 2. A quick internet research showed that UpgradeSummary.wsf script - which I believe previously set this value - was removed in MDT 2013 Update 2 and the value wasn't being set anymore.
I followed up with the user, he added the missing line to SetupComplete.cmd and - as expected - application installation worked on the next feature upgrade installation. Another case closed. I hope that your TechNet troubleshooting met with similar success.
Note: In MDT Build 8443 SetupComplete.cmd already includes the fix described above.