How Long Until I Can Take a Long-haul Flight After an Eyebrow Transplant?

You concern is not a problem, nothing can stop you from flying except the obvious severe surgery complications. Such complications might include bleeding or infection, but it is quite rare  and bleeding might occur in the first day after the hair transplantation. Wear a cool had to minimize any distractions.

"Dr.D"

Flying itself is not a problem unless you were to have a complication of the surgery such as bleeding or infection.  Typically, significant bleeding, is very uncommon and will usually show up in the first 24-48 hours.  Infection typically is apparent in the first week after surgery. Swelling after surgery may be worse in the first few days but in itself isn't a reason not to fly (though if your lids are very swollen, you may need some assistance with vision, e.g. not being able to drive to the airport).  In general if there is a problem, it is most ideal for the surgeon who did the procedure to be in a position to assess the problem.

It therefore is true that a week after the transplant there should be almost no reason not to fly.  You could theoretically fly the next day, though the risk is what's mentioned above.  Every surgeon has different advice. I typically like patients to stay at least the next day to make sure there is no bleeding.