There is a machlokes between the Rabbonon and R. Nosson regarding חי נושא את עצמו. They both agree to the concept when carrying a person, but argue when carrying an animal. If the person or animal is tied down (כפות), the gemara says that even R. Nosson agrees that we don't say חי נושא את עצמו. Rashi (d.h. d'bei) implies that the same logic that the Rabbonon use for why an animal is not included in חי נושא את עצמו which is that it throws itself down rather than assists in being held, R. Nosson applies to something or someone that is tied. Tosafos discusses whether a young child is also considered חי נושא את עצמו and concludes that by a very young child we may not say חי נושא את עצמו (Tosafos implies that R. Nosson who allows for חי נושא את עצמו by an animal, will also say it by a young child, but according to the Rabbonon we won't say חי נושא את עצמו by a very young child). However, Tosafos at the beginning of perek r. eliezer d'mila seems to hold that even on a very young child we say חי נושא את עצמו.
It isn't clear from the gemara what exactly the rationale is to reject the notion of חי נושא את עצמו by a young child. Tosafos raises this question by pointing out that even if we consider a live weight to be lighter, we don't find that there is an exemption for carrying a light weight object on shabbos! Tosafos also rejects the approach that חי נושא את עצמו makes it as if two people are doing the carrying together so that it is considered שנים שעשאוה, because in this case the carrier is יכול whereas the one being carried is אינו יכול so we should consider the יכול to be chayev anyway. Therefore, Tosafos concludes by simply saying it doesn't resemble the carrying that was done in the mishkan. When carrying was done in the mishkan it was on dead animals, not live ones, so carrying is only a violation on dead, not alive. The Korban Nesanel points out that this is a good rationale for R. Nosson who holds of חי נושא את עצמו by animals, but wouldn't be a good rationale for the Rabbonon who make the distinction by people and not by animals? He suggests that the limud from the mishkan is that we don't find that carrying was done on objects that contribute to the carrying by lightening the load, therefore we say חי נושא את עצמו on any situation where the thing being carried lightens its load such as a live person.
There is a famous Radvaz cited by Pischei Teshuva in Yoreh Deiah (195) who permits a Nida and her husband to pass a child to one another based on the sevara of חי נושא את עצמו. In light of Tosafos that this is not a logical distinction, just that it doesn't match the form of carrying that was done in the mishkan, it is very difficult to apply this heter to the harchaka of a Nida passing something to her husband.