Happens constantly, I believe it was Dominos who once lobbied the UK to allow more unskilled migirants in because the UK unskilled workforce basically won't work for them. There will still people willing to do it but not at the same scale. You can see this in other labour markets where migrants are the bulk of the workforce because locals refuse to do that work at that price. Lots of people will choose to be unemployed than work in an unskilled job with unfair wages.