There are many reasons for high-skilled immigrants to support immigration (high-skilled and low-skilled) from their homelands. Loyalty to their fellow homelanders, and a greater interest in the welfare of these, might be a good start. However, there are plenty of selfish reasons as well. More immigration from their homeland would likely mean more people who share the same language and culture, making it possible to gossip in the home language or participate in cultural traditions. It increases the likelihood of the availability of homeland-specific cuisine, and lowers the price of such cuisine. Homeland-specific connections might make it easier to get introductions on jobs or find people to bond with in new and unfamiliar situations.
Armed with knowledge of statistical discrimination (see here and here for starters, or here for a more sophisticated discussion) I came to see that there is a perverse incentive in the opposite direction. Namely, immigrants, like many other people, are often judged statistically based on the average qualities of their “group” which in many cases is all people from their homeland. What this means is that if the average skill level of immigrants from their homeland drops, the immigrant is less likely to be judged as skilled, particularly by strangers. Thus, high-skilled immigrants have a reason to oppose low-skilled immigration from their homeland for the simple reason that it would pull the average down. It’s unclear to me whether this disincentive to support low-skilled immigration would cancel out the pluses.
To make the statistical discrimination analogy clear, consider immigration from Mexico and India to the United States. On average, Mexicans are more, not less, skilled than Indians. However, the stereotype in the US for immigrants from Mexico is that of lower skilled immigrants, whereas Indians are stereotyped as brainy high skilled workers (see here for an anecdotal answer comparing a person’s experience of Indians in the United States versus Indians in India). Why? The reason is that immigration from India is far more selective, with very little illegal immigration, and a much narrower fraction being selected from a considerably larger population. In the case of Mexico, considerable illegal immigration, much of it low-skilled, skews the average skill level downward.
Based on my limited knowledge, I am not aware of any empirical evidence that supports high-skilled immigrants specifically opposing low-skilled immigration from their homelands (as opposed to opposing low-skilled immigration in general). I’m not even sure if this issue has been researched. If, as an empirical matter, they do not, then that probably means that the positive effects far outweigh any worries regarding statistical discrimination.