My impression is that many immigrants see the category "immigrants" as an important group with which they identify, which then gets supplanted in their American-born children by a race-based way of thinking about their minority status. This is anecdotal rather than systematic, but: as a white Anglophone Canadian, I can essentially "pass" totally as a white American, and no American-raised minority person would think of me as anything other than white. But more than once, I have had the experience of chatting with a person who came to this country from Asia as an adult who doesn't know me well, and when I mention that I'm from Canada, they react with a sympathetic "oh, you're an immigrant (or non-American etc.) too", and then become quite willing to share their experiences as an immigrant, including being open about mixed feelings about their and their children's loss of national identity that I suspect they wouldn't be so quick to share with Americans.
If this is right, it may explain in part why Asians have left the Republican party recently, as opposed to a generation ago. I don't think the Republican party's rhetoric about race is any worse now than it was in the days of welfare queens and Willie Horton. But their rhetoric about immigrants has become worse.
(Edit: just to be clear, this is not intended as a complete answer to the question; I don't deny that the points various others have made here are important).
This is an interesting point, as it is true that the Asian solidification as a Democratic voting bloc is a relatively recent phenomenon. As the article notes, Clinton only got 36% of the Asian vote in 1992. In 1996, he lost it to Dole in a much closer showing (48%-44%). Starting in 2000, Gore, Kerry and Obama (twice) have won the Asian vote, with Obama's 73% in 2012 being the high-water mark so far.
In contrast, Latinos as a whole have not ever really supported Republican presidential candidates, at least not in the last 30 years. The Democratic boon/GOP problem there is more that Latino voters as a share/percentage of the total electorate have increased dramatically during that time.
