I'm not on the far right, but I can tell you that of those I know who are, precisely none of them are 'anti-immigrant'. The majority of them would like to reduce illegal immigration and improve border security, but that's also true of the vast majority of Americans as a whole.
Perhaps when you say 'immigrant' you are actually referring to illegal immigrants? Even in that case, over 3/4ths of Republicans favor a pathway to citizenship for existing illegal immigrants (including 'dreamers').
[1] Previous Gallup research has shown that 77% of Americans say that "controlling U.S. borders to halt the flow of illegal immigrants into the U.S." is extremely or very important. And 83% favor "tightening security at U.S. borders." http://news.gallup.com/opinion/polling-matters/209384/buildi...
[2] http://news.gallup.com/poll/193817/republicans-favor-path-ci...
I grew up in Oregon and most of my friends and family have a general disinterest in people moving to the state, especially from California. A neighboring town, often regarded as one of the better towns in the area, has seen drastic immigration (mostly from California) in the past 50 years. On the surface, you might not see much damage, and in fact you might see only improvements. But when you look at the side effects, such as multi-generational families getting kicked out due to excessively high taxes brought by wealthy Californians, and general road traffic increasing in speed beyond what locals want (Oregon doesn't have any speed limits above 65mph, AFAICR), you see another story.
Immigration is great when it works out for you, and not so great when it doesn't.
Edit: I guess the broader question is to what degree do locals get to infringe on the rights of outsiders to buy property?
Its pretty anti-immigrant. Everyone is progressive with other people's pockets.