HR out-sourcing was one of the earliest, though it was mostly domestic (i.e. on-shore).

Ditto accounting: the Big 3 or Big 6 or whatever the current label is. I recall trepidation when the B-school at the U solicited funds from them for remodeling, and installed brass plaques outside each class-room door "A Accounting Bodyshop", "B & C Accounting Bodyshop", "D Accounting Bodyshop"...

Out-source firms that specialize in pay-roll processing -- just one sub-set of accounting -- have been around for quite a while. One of the last identifiable fragments of the super-computer/financial services/software products/custom programming and analysis services firm where I once worked is the pay-roll processing, back then a teeny tiny sub-sub-subsidiary... but, tellingly, it was the one the new company CEO hitched his wagon to and rode down the cliff as the sequence of the destroyed a GREAT AMERICAN TECH firm.

Execs of non-tech firms have always considered computing services ("data processing", or "IT" as most folks at Tech Republic and Ziff Davis apparently think of it) to be a cost center. But they expected it to reduce their costs, and it did, but never as much as they wished, because, quite frankly, they never really understood computing, never wanted to invest the effort to do so.