In my practice, lower eyelid blepharoplasty is approached via a transconjunctival incision, meaning that the cut is made on the inside of the lid, where it can easily be concealed. This approach is also safer because it protects the septum, which, when violated through the skin, can increase the likelihood of the lower eyelid being pulled down or retracted.
For skin removal in cases where there is skin laxity, the skin is tightened from the outside at the outer corner of the lower eyelid, but the incision hides nicely in the natural skin line. Also, because only skin is removed through that cut, the likelihood of the eyelid being pulled down is significantly reduced because the septum is not violated. The lower eyelid blepharoplasty procedure is typically performed with a combination of local anesthesia through injections of nerve block in the area as well as either IV sedation or full general anesthesia, though these days, most patients opt for IV sedation to reduce the amount of sedation they receive.