Sure, look at the Hmong in the US. Poor and rural before they came to the US, and still, on average, pretty poor and doing pretty badly in the US.

Meanwhile, I have a lot of Chinese, Hongkongese, and Korean coworkers whose families are well-off enough that they don't need to work for a living, have relatives who don't work for a living, etc.

So yes, lots of first-generation immigrants in US come from serious money overseas, and the ones who don't have worse outcomes on average.