Yes. Just as the new kids on the block, or the new kids in a school, always have to prove themselves before being fully accepted, so it is with immigrants. But that does not mean that all immigrants are equally congenial; that all immigration is equally congruous to the traditional American culture.
The iteration of the initial rejection of Irish, or Jewish or other 19th Century immigrants, has been used consistently, generations after they finally were accepted, to increase the Leftwing vote among groups whose interests are manifestly being betrayed by those Leftist demagogues, seeking to fragment society, and deprive all of us of fundamental liberties.
The Irish and the English, Scots, etc., are not that different in relation to their purpose in coming here. That is reality. The hordes pouring over the open border, may include a few with similar purpose; but that is not the norm among them. The perceived America that draws the present crowd, is the political climate that is strangling those of us who adhere to the philosophy of the Founding Fathers.
You cannot equate the two streams--even if you choose to ignore the fact that we are already terribly over-congested in urban areas. From a long term social standpoint, we do not need millions of new arrivals competing for the same resources, space, etc., while in their very congestion increasing the likelihood both of medical epidemics & civil unrest.
We desperately need a frank discussion of all relevant issues; not the 'politically correct' fear of walking on metaphorical "egg shells."