I have a whole bunch of cards I created with the wrong work item type. I read Damian’s post on how to do this. When I performed the steps in the TFS Web Access and in Visual Studio the existing links did not stay linked to the copy.
In Damian’s post he states that it is a deep copy of the work item (and he even has screen shots.) So at this point I am not sure if it is because of 2013 Update 4 or if the change in the work item type is causing this.
Below is a copy where you can see only the related link exists.
These are custom WIT but I have also tried it on Features, Stories, etc..