As part of every ashkenazi brit milah ceremony I have attended, the baby was passed from a woman (the kvatterin) to a man (the kvatter, always her husband?) who then brought the baby in to the synagogue, before the baby is given to the baby's father (and from him to someone else who placed the baby on the kisei shel eliyahu).
What sources discuss this honour of kvatter/in? When does it seem to have started?
I am aware of Rema in Yoreh De'ah 265:11 who writes that a woman brings the baby to the synagogue, and then hands him to her husband who is the sandak. When did this intermediate stage of a kvatter taking the baby from his wife (rather than the sandak's wife) and taking the baby towards the sandak begin?