I'm skeptical that the GOP can appeal to immigrants like the Canadian Tories or NZ's National Party can. The immigrants to both of those countries have a much higher % of "natural conservatives" since they are more Asian and have higher incomes compared to the largely Hispanic immigrants to America.
It's worth comparing America's Asian populations to Canada's.
A lot of Asian-Americans are not first-, second- or even third-generation immigrants. Their ancestors came here in the 19th and early 20th century to perform menial work building railroads and doing laundry. Their economic ascendancy was analogous to that of groups like the Irish, but unlike "white ethnics" they, perhaps because of their obviously non-Caucasian physical appearance and non-Christianity, did not socially "become white" the way Irish, Italians and Poles gradually did.
The relatively recent Asian immigrants are arguably more receptive to the Republican Party. They didn't come here to do grunt work like the 19th century immigrants did. They're either highly educated "model minorities" or they're coming to join family that already lives here and often are successful business owners. And many of them are Christian - particularly the Korean-Americans.
Immigrants from the Indian Subcontinent are a harder sell for Republicans. Many of them grew up in India at a time when it was very socialist but also a democracy, so they don't associate Leftist economics with oppression the way Russian, European or East Asian immigrants might. And when you are from or spent a considerable amount of time in a country where poverty is commonplace and very visible, it's hard not to believe the government ought to do more to help people out of poverty. And a culture that still has a high degree of socioeconomic determinism via the caste system probably isn't going to buy the Republican mantra of, "Anyone can become a multimillionaire and if they're poor it's because they're lazy and not trying hard enough."
