I was a part of a few efforts to offshore some technical support type work across a couple companies. It was highly technical equipment so it wasn't your typical customer service or base level PC support, at the same time there was no reason it shouldn't have worked to offshore.
Every effort failed miserably.
The core of the issue was getting / maintaining talent / a real lack of professionalism in the folks we ended up with / lack of dedication to the job.
If there was any reasonable amount of saving money per workload, you just couldn't get a company to outsource to who actually kept their own workers around long enough to be productive. Anyone who was productive almost immediately proved their worth and left for a better job, or as we suspect the outsourcing companies actually moved them to clients that paid more (in the case of us hiring them directly, we knew they left).
The folks you did get would close cases randomly / played a sort of hardball metrics game where they really just upset customers / state side we'd end up cleaning up the mess. Lots of technically correct solutions / statements that really don't solve the problem... And in the meantime it all ate up a TON of upper and lower management time trying to get things right / set up... meanwhile there was a US, a European, and even in one case a Japan side support team that needed little interaction from management.
Another effort tried to establish a local presence offshore (a few companies already had resources there) ... same results.
The end result was all the effort and costs resulted in it being more expensive to offshore in many cases if you looked at the real numbers.
To be clear I'm sure there are plenty of professional folks out there to be hired (we had plenty of those folks employed FROM those places who were outstanding at their job) I suspect the larger issue was that the savings weren't really that apparent, and what quality / dedication you get for X price isn't always as apparent when someone says "Hey X resource is Y cheaper over there."
Quality and outcomes are hard to calculate / something being cheaper on a spreadsheet doesn't mean it really is.