The psychological motive to abduct a child is likely a result of the maternal instinct of women. If a woman has a miscarriage or discovers that she is barren, she may feel empty and try to fill the void with a child. The act of abducting another mother's baby is not well-researched, but is likely a deviant act resulting from the pressures of not being able to have her own baby.
Historically speaking, society has placed norms on women to be mothers and to spend much of their life rearing and taking care of their own children. Abductions made to obtain a child to rear have both psychological and sociological implications that extend beyond normal criminology studies. This fact, combined with the rarity of such cases, leaves the subject relatively mysterious.
What we do know, however, is that the crime of child abductions for child rearing is often premeditated. This means that women consciously think about and decide to commit the crime, furthering the psychological aspect of the crime. Maternity, after all, is not as common a motive to commit a crime as money or fame. The numbers of children living in these circumstances are unknown, as it is almost impossible to identify a child that is growing up, especially if he or she was abducted at a young age.