I have experienced this a lot. My parents immigrated from Portugal, and Italy.
We have been back, and I often have that feeling of connection to the culture, the people and the land. It helps that I am first generation born here, so the family culture we have here is similar to the culture back in Europe.
However, what I never forget is that the Europe they left is not the Europe that exists now. Portugal was under a dictatorship when my mother left. There was zero opportunity.
My grandfather left Italy in the 1950s, after over a decade of hardship for him (he was in the Italian Army). There was huge migration happening in Italy at the time, from South to north, of people seeking new opportunities. Whole communities in central and southern Italy were gutted. I think something like 25% of families were living in poverty. A huge economic boom eventually happened in the 1960s, but for my grandfather and other people at the time, given what was happening around them, that was probably not a obvious conclusion.