Agree 100%. If they onshore it to MindPearl, for example, with the same training and level of authority, then I am not sure support is going to be any better. If they bring the call centres inhouse then there's a chance.
My experience with three companies with internal support (IT, accounts payable, HR etc) that was "offshored" was that support was 300% better in the two companies that "offshored" to fully fledged company employees (for example working for the same company as me, but in the Philippines) than in the one company that offshored to non-employees working for a service company. Employees received better training, were given more responsibility and shared in the overall success of the company, and not just driven by churn and number of tickets closed.