I always recommend my patient to be active after Brazilian butt lift. You should stay in the area where you have your procedure done for at least 7 days to make sure that everything is going well, your pain is well controlled and that you are in good condition to be on a flight for such a long period of time. Clots do not typically occur on healthy patient after surgery unless the patient was not active after the procedure. Immobility after surgery increases the risk of clots after surgery due to pooling of the blood in the larger vessels of your leg setting the stage for a cascade of event that result in clots. Most patients after a Brazilian butt lift do not complain of significant pain. If your pain threshold is low then this needs to be address before your flight so you can move around without significant problems.
Anticoagulation in a healthy patient is currently controversial. There is consensus by most surgeons that patient with high risk for clots should have anticoagulation prophylaxis. On the other hand, cosmetic patients usually do not have significant risk factors to justify anticoagulation. Nevertheless, I sometime use these medications (Lovenox) on a patient to patient basis. For example, if I notice that the patient is not very active, pain threshold is low preventing the patient from moving, long flight after surgery etc, I will suggest anticoagulation. Remember it can be a catch 22 because anticoagulation can prevent clots but also increase the chances of bleeding. In addition, anticoagulation needs to be done for two weeks after surgery since it is well know that you can develop a clot even after 10-14 days after surgery.