Because their income is discounted, they are more likely to receive social assistance and they send money abroad, in the short-medium term their economic effect is similar to offshoring.
Long term, they bring their family over, raise families in the host country, get better jobs and integrate (culturally and economically) so the offshoring effect disappears.
The main problem is, especially concerning the rise of populism, when the short term effects become too severe, the long term outcomes don't matter to those who have been affected by lowering wages during their working years. And it's not every industry, perhaps not even a majority who are affected, but for those who are, they feel as though their social contract has been broken, and they vote for whoever promises to fix it.
This forum is mostly IT workers. For awhile, wages in this sector in the US were suppressed by things like cartels and H1B visas, and subsequently action was taken to enforce proper employment laws and increase the wage requirements for H1Bs. That was 1 sector, not even that much immigration, and wages showed a noticeable dip. People here were outraged too, but it was before the Trump years so the whole conversation had a different tone.
Anyhow, I don't think immigration is bad, but there are pros and cons, different effects on the economy and people, and these definitely need to be accounted for.