Bait and Switch
Anti-immigration activists frequently claim to be concerned about the alleged problem of low-IQ immigrants. You can see examples in the comments to Bryan Caplan's recommendation to offset low-IQ immigrants with high-IQ immigrants. The only problem is that they are equally offended by high-IQ immigrants. For example, the proposed immigration bill is designed to allow in more skilled immigrants but some allegedly-sane people are frothing at the mouth about it. It looks like the anti-immigration people set out the bait of “We only want the best people.” but switch to “Keep out all foreigners!” once that's accepted.