Is it permissible for a Muslim woman to breastfeed a Christian child, or for a Christian woman to breastfeed a Muslim child? And what is the jurisprudential ruling for the child after breastfeeding?

It is permissible for a Muslim woman to breastfeed a Christian child, and for a Christian woman to breastfeed a Muslim child, as the basic principle in this regard is permissibility, and it falls under the category of showing kindness. The rulings of Islam related to them do not change due to breastfeeding.