If you go back far enough, you can get to a time when the Republican party pro-business and anti-government spending, but not necessarily more (and generally less) pro-white supremacy than the Democratic party. And they were obviously successful at that point and won national elections. However, here's Grover Cleveland, the only Democrat elected president between the Civil War and 1913, talking about immigration in 1897:
A century's stupendous growth, largely due to the assimilation and thrift of millions of sturdy and patriotic adopted citizens, attests the success of this generous and free-handed policy which, while guarding the people's interests, exacts from our immigrants only physical and moral soundness and a willingness and ability to work.Here's Woodrow Wilson on the subject in 1915:
Restrictions like these, adopted earlier in our history as a Nation, would very materially have altered the course and cooled the humane ardors of our politics. The right of political asylum has brought to this country many a man of noble character and elevated purpose who was marked as an outlaw in his own less fortunate land.And here's Henry Cabot Lodge, one of the most prominent Republican senators of the time, talking about immigration in 1910:
There is a growing and constantly active demand for more restrictive legislation. This demand rests on two grounds, both equally important. One is the effect upon the quality of our citizenship caused by the rapid introduction of this vast and practically unrestricted immigration, and the other, the effect of this immigration upon rates of wages and the standard of living among our working people.
(all quotes found here)
Notice a difference? Presidents Harding and Coolidge both passed legislation restricting the immigration of Southern and Eastern Europeans. Hoover actually tried to suspend all immigration during the Great Depression. I honestly don't know how much nativism has always been a portion of the Republican appeal (although it might be telling that Catholics used to be as uniformly Democrat as blacks were Republican in the nineteen-teens and twenties), but it does kind of seem like it wasn't all austerity and fiscal discipline back then, either. It'd be mistake to assume that the Republican party is over and done with just because they only managed 48% of the popular vote this time around, but I think it's also a mistake to assume that a simple shift on immigration will actually be all that simple.