That last part of "demanding more pay" almost never happens. The reality is, service economies start a race to the bottom by taking advantage of the lower starting cost of living and then the economy doesn't really improve. People just live with lower starting costs of living and lower quality of life in many cases.
Note that I use the term lower starting costs of living. There's a reason a lot of people migrate if they can. It's because these countries (like Sri Lanka which is where I live) have costs that suddenly spike the moment you aspire to a better quality of life and the salaries don't keep up.
But I digress. The evidence of the local economies not necessarily improving is there in many outsourced service economy based countries. India is a prime example. And when outsourced jobs started moving away, there were articles describing India's IT industry as a bloodbath (or something similar).
So no. Paying people better actually improves the economy of the country. Especially if it's non local currency. Most countries benefit from inflows of strong foreign currency. People start having less incentive to leave the country so brain drain is less. People can reach higher quality life goals, so purchasing and usage of local services increase (think food and beverage, local tourism, clothing, and even other simple services like laundry).
The idea that US companies paying higher rates is going to wreck an economy of a lower performing country is very unlikely. The reality is that if it somehow became a norm, (unlikely since if rates are high globally, you'd keep the jobs local as much as possible), people would just form new companies to service that market.
Which is technically what is already happening in countries like India and Sri Lanka.
So no. Please pay people better. It'll most likely help that country's economy.