Appellate courts may notice when a brief is recycled. I imagine that usually some staff attorney notices these kinds of thing. When such a thing is found though, the court will not hide its fury. The federal Ninth Circuit didn't in this case recently blogged about. People who live in glass houses, though, should not throw stones. A recent scholarly article points out that appeals courts' recycling nonprecedential opinions can give that recycled material greater influence than precedents.
That both of these issues arise as to immigration appeals is not accidental. I don't remember which immigration lawyer told me that much of immigration advocacy is to lose very slowly. Waiting lists in administrative processes can be more than 10 years long. In immigration proceedings, every additional hearing, review or appeal can slow an unwanted action or loss of a desirable status for months or years.
It should be no surprise that it is common for institutions in this system are crammed with cases, not infrequently of dubious merit. Client's counsels' offices sometimes see computer cut-and-paste briefs as an easy, cheap to do, yet large fee-bearing way to slow down bad results. Similarly, federal courts of appeals justices commonly refer the flood of appeals of Board of Immigration Appeals decisions to court staff attorneys who are expected to churn and burn through the load. They end up cutting and pasting copy from old non-precedential opinions into later ones. This copy may get outdated, but it may take a while before someone with the power to fix it may notice, viz. an appeals court justice, and by then, many cases may have been irretrievably wrongfully decided
Sometimes cases that do not have any outwardly visible immigration issues go through appeals because there is a collateral immigration benefit to someone associated with one of the parties for the case to keep pending.