Whether redheads will go extinct or not will be determined not by blood-mixing, but by fashion and cultural preferences. By 2060 genetic engineering will have become so commonplace that parents will be able to choose certain traits in their still-to-be-born kids.
Blond hair and blue eyes will probably start popping up a dime a dozen among babies born in America or Canada in the decade of the 2060's, because in North American culture they are highly valued. The phenomenon of race itself will undergo selective modification. Large numbers of white people will get their genes modified so they can be very light in winter, and yet grow a deep and rich tan in summer. Meanwhile Asians will seek eyes that are less epicanthic, and Hispanics and persons of East Indian descent will make their children somewhat lighter.
In a nutshell whites will get sometimes closer to Arabs or Mexicans, while Arabs, Mexicans and East Indians will grow more in the direction of whites. What blacks will do, I cannot foresee so easily. Some may opt for more Caucasoid traits, others seek to entrench their negroid details.
At some point, government will need to step in and legislate certain clear limits. "The best interests of the unborn gene-engineered" will be the criterion. And you'll get the religious element screaming: "No,no! This is sacrilege! You are interfering with God's creation."
The true criterion of what is allowed or not (in genetic engineering ) should be not whatever anyone believes is Heaven's will, but whether changes enhance or worsen THE QUALITY OF LIFE for the genetically modified. Removal of genes that cause diseases or defects, for example, is a no-brainer. But life is something sacred, and we cannot have people altering it at whim or for crass commercial gain. This decade everyone wants redheads, so we have a spate of redheads. Twenty years from now, they all want blonds....hey, wait a minute!
What really counts here --- the fancy of the parents, or the best interests of the child to come?