No, you need to study up a bit. Migration and mixing may reduce the incidence of gene expression, but will not reduce the number of carriers for the recessive genes. Cystic fibrosis is also an autosomal recessive trait and the individuals with the disease are frequently infertile. Despite the inability of the homozygous individuals to reproduce, the recessive allele has not "died out". This story did not pass the freshman biology smell test.
If the group that carries blonde genes is not reproducing at replacement levels, how is it that the gene will not die out?
Granted, there are small populations of Caucasians who are having more than 2 children, but these groups, usually evangelical Christians, are small in number. However, other ethnic groups that do not carry blonde genes are reproducing in large numbers and doing well.
I have monitored several of these threads and seen a lot of joking and some short-tempered chastisement of those of us who "cannot pass freshmen biology." What I haven't seen is an honest analysis and description of how population genetics work, in a way that demonstrates this purported phenomenon is bogus. Please help us to understand how this allele will not die out.