Brahmins in India had high social status but less money/assets in past. So I am not talking about economic status.
And indeed, Mexico is gaining ground on the US, not least because of bilateral trade and bilateral migration.
Back then labor was in more demand. Evidence is the salves were imported often at great cost and risk.
There were fewer commodities in market.
Germany had coal, French needed it.
There was no limit on immigration.
People from Germany and France were more equal in education and social status than the difference you've between Americans and Mexicans.
Technology wise countries in Europe were quite equal, what differed is the amount of resources at their disposal.
>So it plainly does not hold that trade and migration entirely eliminates differences.
It doesn't have to entirely eliminate differences, just have to make them more equal.
> People from Germany and France were more equal in education and social status than the difference you've between Americans and Mexicans.
Good way to disqualify any counter-examples to your theory. If they're more equal in status (i.e. have more equal economies), then they don't count.
Look at the US and Mexico: Despite having a lot of trade and migration, the two countries significantly differ economically.
So it plainly does not hold that trade and migration entirely eliminates differences.