I don't think most people have a problem with immigration. A term that gets used to cover many things these days.
If you listen closely, you will find that people don't like two things. 1) Mass immigration (we probably need to further define that). And illegal immigration. I think most people understand (at least in America) that we are founded on immigration, and we enjoy amazing food, and cultures because of it.
Mass immigration - large number of people immigrating and displacing the existing population within a short period of time (probably also needs defined). I think when places and areas change slowly it is much smoother. Its like boiling a frog. Regardless of where you are from it can be upsetting to see your culture displaced by another. And that is why we have limits on how many people can immigrate to America every year. Some say they are too high, some say they are too low. I am sure we will find just the right amount one day.
As for a short period of time - that is mostly defined in generations. For immigration to work I think it needs to happen slow. Both sides need to adjust. The existing culture needs time to meld with the arriving culture, and the arriving culture needs time to adjust to the new culture. Both sides will need to be willing to give and take to form a NEW culture. Neither side can force full respect of the entirety of all parts of either culture. Doing so will result in generational disputes and battles.
I think most things can generally be solved with culture clash by having a potluck and enjoying each others food. Fat bellies of tasty food can bring people from all walks of life together.