Does the Nose Tip Go Down Years After Surgery?

 All tips drop in the first 12 months after surgery. All noses drop after 30-50 years of aging. Do not have the doctor elevate the tip for aging decades later! A good rhinoplasty surgeon won't do this anyway.

Most of the healing after rhinoplasty occurs during the first year. There are subtle changes that occur over the next few years. However, the nose also ages. Like all parts of the body, as we age, the nose will droop. Ligaments stretch and skin sags.
In rhinoplasty we perform surgical maneuvers in anticipation of what the nose should look like approximately one year from surgery. So, I suggest you find a rhinoplasty expert and let them plan surgery as they see fit for your nose.

With the passage of enough time, yes, your nasal tip will droop from loss of support. However, in my opinion, this is not a reason to have it overcorrected in the short term.