When deciding to outsorce the work actually begins
No one can take over the support without someone will notice. In the beginning, response time will be slower and perhaps not accurate (because of the person handling the case do not know your environment)
depending on the size of your server-farm, this will take some time to let the people get "into it".
There's also the aspect of documentation, did you hand over any type of documentation regarding your environment or do the new support have to guess what goes where and which systems are connected in what way?
My dayjob is in the "support" business where a large part is to handle different companys servers and their application. I generally do not handle anything INSIDE their business systems. If they can connect, my work is done. Someone else have to take the support on that part.
I do have colleagues that takes 1-st line support, with questions like "how do I add a signature to my email software", if there is anything more technical, they flag a case to me and I make the call what to do with the problem.
What you always need is someone that takes the command and feel responsible for the customers situation.
And ofcourse do anything to solve the customers problems.