Migrants & immigrants often care about what happens in & to the countries of their origins. This has been always true throughout the history of mass migration under capitalism, but with the greater ease of travel & communication today, political powers of migrants & immigrants will have more visible & complicated impacts on both the USA & the rest of the world than before, both for better and worse.
Quantitatively, immigrants' impact on U.S. politics is probably smaller today than early in the twentieth century, in that back then the ratio of the foreign-born to the native-born was higher than now. Qualitatively, however, the prominence of immigrants of color from Asia & Latin America changes the politics of race in a way that is difficult to calculate. A lot depends upon whether immigrants of color will make a political alliance with working-class blacks.
Migration & immigration to the USA (& other rich nations) may affect the rest of the world more significantly than the USA itself. For instance, remittances from migrants & immigrants help support the economies of many nations. At the same time, brain drains are economic afflictions upon poor nations, who end up wasting their educational investment. Moreover, emigration works as a safety valve to diminish political discontent & social turmoil in poor nations. Blocked ambitions of the educated & energetic (who many migrants & immigrants are) are often more politically dangerous than unrelieved miseries of the most beaten down. Perhaps most importantly, migrants & immigrants of every political stripe -- left, right, whatever -- get themselves involved in trans-national political endeavors: fundraising, lobbying, recruiting, networking, propagandizing, and so forth, including returning to their countries as politicians, economic experts, & sometimes as guerilla warriors.
Yoshie