It'll never die out but what you'll find is that there will be significant decrease in marriages between first cousins, and there will be an overall decrease in marriages between people who are relatives.
For those unfamiliar with why it is so common amongst British Pakistani's here is a bit of background.
1. Pakistani society on the whole has largely been tribal, particularly in rural areas. Marriages in such cultures have predominantly been arranged and typically been within the community of people you know or have some sort of link to. In rural societies these links aren't particular far; in a geographic sense.
2. When Pakistani's migrated to Britain, they were predominantly from these rural areas. These people took their culture with them. When they had children they wanted to marry them into the communities they knew, typically these ere communities back in Pakistan.
3. At the same time poor people have found that marriage was a way of getting someone from their family to the UK, so they could also earn and send money home, strengthening the financial position of their close family. In areas like AJK, if it wasn't for expat money, we'd be living in really poor conditions.
This led to a culture being established where a lot of young people were co-erced into marrying their first cousins, under the guise of "these are the decent matches we can find for you" but in reality there was a strong dose of "it also meants my brother or sister can live the rest of their lives financially comfortably". At the same time it was also a lazy way of finding marriage partners for all involved, low risk because they knew the character of the parties involved.
This culture is dying out for several reasons;
1. The genetic risks, particularly of repeated marriage to close relatives. People are aware of this and very wary of this. Even in Pakistan people are doing genetic tests before getting married.
2. The social drawbacks. First generation and to an extent second generation immigrants weren't culturally British. We're now in our third and fourth generation. These kids speak English and only know "ammi abbu" urdu/punjabi/mirpuri. When they marry people from a culture they're not themselves immersed in, marriage becomes a bit more difficult.
3. Dealing with freshies. A lot of them from the pind have no life experience and you spend forever hand holding them through the norms of society. This is particularly difficult if you marry one of those mooch twisting ultra macho types with no real substance who feel they've been kicked in the nuts every time a woman corrects them. The adaptation period is hard, they never really get your jokes, you never really get to watch the same TV etc.
4. The British government has raised standards for spouse visas. You need to be able to speak english, your sponsor needs to be earning £21,000 a year (not a lot but above minimum wage). A lot of the community don't declare they earn that kind of cash. A lot of the pendu's can't pass the tests.
5. Clash of expectations. A lot of people a few generations ago were manual labourers, the economic disparity between those of our family in the UK and back home might have been high, but the difference is lifestyle wasn't all that much different. Today more and more boys and girls especially are educated career types. If you're making £30-40,000 a year as a professional, you don't want to give that up to live off taxi driver wages, spending your day cleaning and watching TV. The alternative often is having to work and clean/cook with no support in household chores from your FOB husband.
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I married a distant cousin from Pakistan. We're happily married but I think it helped that I was emmersed in Pakistani culture, fluent in our mirpuri language, and we had shared values. My wife wasn't from a pind, she worked in Pakistan, she had her own small business yet, when she came here her English was quite poor and her confidence suprisingly low. It took ages to get her to be independent, to research before asking questions, to try talking to white people, rather than looking at me to answer the questions addressed to her. Alhamdulillah she's adapted, she's now got qualifications, got a job, does the kids homework.
Marriage is a partnership, everyone needs to be flexible. Cousin marriages won't result in deformed babies,but can sharply increase the risk, especially through multiple generations. I still advice everyone to get to know thefamilies you marry into before you get married, we're Pakistani, we never quite marry 1 person.