A plastic surgeon versus dentist versus aesthetic doctor versus nurse is a difficult comparison to make, because it very much depends on the individual's training, experience, and expertise. I think the most important thing is to make sure that the practitioner is medically qualified (whether as a dentist, a surgeon, an aesthetic doctor, or nurse), fully insured, and properly trained. There are excellent, highly expert injectors who are not plastic surgeons but may be dentists or nurses. Make sure whoever you choose has the expertise you're looking for - it's good to research whoever you're going to and perfectly fair to ask how much experience they've had, and ask for pictures of previous work. If you're satisfied that they know what they're doing, and feel that you're happy with the consultation, absolutely feel comfortable booking a treatment!
Lip filler migration occurs when it is not properly placed. It is due to technique, not to any particular product. This is something that can be best avoided by visiting an experienced practitioner who has a good understanding of the facial anatomy.