Nasal tip depression while talking. Why is this happening, and how can it be corrected?

Your nose is coming down during lip movement and smiling because of two issues: 1. The depressor septi muscle pulls down your nose durin these movements. It can be resected. 2.  Nasal tip support. It can be strengthened with grafting. Both these issues can be addressed endonasally or through an open approach, depending on whether is any other issues that need to be addressed. 

This is usually due to the depressor septi muscle. This can be treated either with Botox for a temporary, non-surgical solution or the muscle can be surgically removed and structure added to support the tip. I hope this information is helpful for you.Stephen Weber MD, FACSDenver Facial Plastic Surgeon

The muscle can be released. I perform these surgeries with closed technique
to preserve the structure of the nose and minimize visible scarring.Kenneth Hughes, MDLos Angeles, CA

Some patients have a prominent depressor septi ligament which pulls the tip down when smiling. The depressor septi ligament is attached to the smile muscle and pulls the tip down with  animation and is especially prominent in a down-turned nose. This ligament can be released at the time of the rhinoplasty procedure. Digital computer imaging would be helpful understand what can be accomplished with your new nose upon your facial features. For many examples, please see the link below

My response is based only on your description - pictures would be helpful. This movement is probably secondary to contraction of the depressor muscle which is located at the junction of the top of your lip and the base of your nose; this can be cut.