Can fillers last up to 5 years?

Yes, some fillers, like Bellafill, can last at least five years. Whether or not they can migrate probably depends on how injected but I do not think that that is common. I do not believe that only 1cc of filler should be injected at once as it can take multiple syringes of filler to achieve a noticeable or good result. If injected properly there is no specific limit. I would say that in many cases(not including lips) 1cc of filler would not make much difference and would leave a patient unsatisfied with results. 1cc might do well if targeting a specific problem in limited area (example:marionette lines). Marionette fillers do well if  correct filler chosen and injected properly. Be sure that your plastic surgeon or  dermatologist has experience in injecting fillers before choosing them. In addition, you might see  fillers long after injected when MRI or biopsy  done but that does not mean that enough is still present to correct the patients problem so be aware of answers that you get and know difference between filler being there in 2 or 5 years vs. enough filler for desired  results being there in 2-5 years. 

Different fillers last different lengths of time, depending on their properties of degradation and the individual patient's body chemistry with respect to that particular filler.  The same is true of a filler's propensity to migrate: its likelihood is variable depending on the product, where it was injected, and the individual patient's body chemistry response to that filler.
In general, the hyaluronic acids break down fastest, and the more highly-cross-linked hyaluronic acids tend to break down slower than the less-cross-linked ones. The hyaluronic acids act as a sponge, taking on water and swelling, to fill in a defect or hollow. That is their mechanism of action.  The only collagen production is due to the needle injury from the needle used to inject them. 
The 3 most common Biostimulatory agents injected into the face in the USA act by triggering 2 kinds of collagen (each). They include Radiesse, Sculptra, and BellaFill.  Part of the volumization and improvement in skin texture is, therefore, related to the body's response to the biostimulatory agent: Collagen production!  Radiesse (Calcium hydroxylapatite), a biostimulatory agent, tends to last longer than the hyaluronic acids, and triggers synthesis of two types of collagen, and actually allows collagen and muscle fibers to integrate and grow into the spherules of product.  Sculptra (polylactic acid- which is a liquified, crystallized form of a commonly used dissolvable surgical suture material), another biostimulatory agent, acts longer still, so long as the product is injected in a series of sessions (much like a vaccine).  And BellaFill, (PMMA, poly-methyl-methacrylate), which comes suspended in a medium that contains bovine protein, lasts forever!!!  PMMA is a material with an excellent track record, incidentally, for its safety and biocompatibility, having been used as a material to replace the eye lens (after cataract removal) for generations!!
As we say, you can move back the hands of time, but you cannot stop them!  So with all these products, signs of aging will eventually show, since aging is a continuous process.
Thank you for the interesting notes you shared!  I love the science of all this, and the plethora of wonderful ways in which we can apply it to help others.

Even hyaluronic acid fillers which are supposed to last only about 6 months or so can last an eternity in some patients. Those patients bodies simply do not see the filler as a foreign object and no response is built against it. Unfortunately you did not mention the fillers you were talking about. some fillers can move but only one injected improperly by the surgeon or whoever the injector is. In the majority of cases when done properly the filler should not move filler it is a known fact at least to me and without an MRI that fillers in the lower eyelid will move almost in every one as there's simply not enough connective tissue to hold the filler in place and the arby calories accurately muscle tends to squeeze it into a place where there's absolutely no resistance, with that place being the area between the eye and the nose in the lower eyelid area. I hope this helps and good luck from New York City