Can non-surgical nose fillers migrate over time?

You are asking an excellent question.  I have clients who present to me for non surgical rhinoplasty who have been injected in the past.  If they have been injected with a hyaluronic acid product such as Restylane or Restylane Lyft, then I use hyaluronidase to dissolve the filler in the nose prior to performing any further injections.  I have the client come back to my office a week later and we essentially start with a fresh slate.  Otherwise you will get filler which is at different stages of dissolving and that will lead to irregularities in the nose over time.

Widening of the nose after injectable fillers is more likely due to over correction, rather than migration. Thank you for your question. Sincerely, Dr. Joseph

When filler injections are performed by a trained expert like a double board-certified facial plastic surgeon, the risks of complications are significantly diminished. Injectors with this degree of training understand the layers of the skin and how to properly place needles or cannulae for maximum effects without affecting blood vessels or causing lumpy areas to form. 
Fillers expand somewhat after injection, as Hyaluronic Acid gel absorbs water and will smooth out as it swells. This isn't exactly migration, as you mentioned, but it does explain minor changes especially during the first 2 weeks following injections.
I have never encountered a case of filler migration that would cause a deformity making a client look like an "avatar" character, but if this is a concern you have, you should make sure that your injector is properly trained in filler injections, and that they are an expert in nasal anatomy. 
I suggest having a double board-certified Facial Plastic and Reconstructive surgeon & E.N.T. for all nasal cosmetic procedures.