Being American is an accident
Where you're born is an accident. True. Not fair, but also not something that can be controlled. It is not the duty of the leaders of wealthy nations to correct global inequality. Their primary obligation should be to protect the interests of their constituents. The onus should be on other poorer nations to raise their labor standards to a higher level.
There have been historical injustices, but this excuse cannot be used forever. The United States, Canada, Western Europe, Australia and Japan (among others) have shown the model for relative prosperity, peace and freedom. It isn't perfect, but it's the best the world has seen. The governments of third world nations need to follow this blueprint. The answer is not to import unlimited numbers of third world migrants into the the first world. Those countries must be improved from within, and then all these problems would go away. In that scenario, we could have something more closely resembling mutually beneficial fair trade and a vibrant global economy.
The question that needs to be asked is what is preventing these third world countries from joining the first world? I agree that some wealth redistribution is necessary. The crucial point is where the wealth is going. That's why labor standards need to be raised for workers in other countries. When jobs are shipped overseas, I want a good part of that money going to the foreign worker to raise their standard of living. It's absolutely appalling (albeit completely rational and predictable) for corporations to exploit cheap foreign labor. The big loser is the American (or UK) worker, the smaller winner is the foreign worker (although their working conditions will probably be deplorable), the big winner is the corporation and to a lesser extent the consumer. The only way this gap is ever going to close and this dilemma get solved is for the third world to join the 1st world. I just don't see it happening any time soon.
In the end, this blurb is the crux of the disagreement:
American income inequality is bad because it is regressing to the global mean. The global mean is the result of a system largely shaped by America and the actions of the American people. The correct approach, now that we're feeling the sting, is not to prevent the regression and prop up the system, it's to coordinate and fix the mean so that the regression doesn't hurt so much.
Embedded in this paragraph is the assumption that the current 1st world nations are at fault for the state of the third world today. In other words, the United States, Europe, Canada, Japan etc. are successful because they took advantage of the third world and conversely the third world's state of relative poverty and turmoil is due to being exploited. Therefore, the first world has a moral obligation to make reparations and level the playing field.
Certainly there is an element of truth in this evaluation, but at a certain point, the poorer nations need to be held accountable for their own self determination. They are at a historical disadvantage due to the actions of our ancestors, but those alive today are not responsible for those actions. The wealthy countries have shown the blue print. These other countries need to adopt the institutions of that we know work (democracy, free market, free speech, separation of church and state, and so on). But there seems to be so many barriers (corruption, instability, ethnic and religious conflict, extreme poverty and so on). that I'm not optimistic this will happen any time soon. Whenever the west intervenes, it rarely goes well and they tend to get criticized for meddling and imperialism. However, any calls for pull pack or withdrawal from the global scene are inevitably met with derision and accusations of "isolationism" and/or callous indifference. It's a no win situation.