If I recall correctly, I think someone with experience had mentioned that women who regularly partake of kava will cease intake when they discover they are pregnant. Lebot mentions in Tanna women may use a leaf extraction close to their delivery date to help with presentation. He also mentions that the leaves have been used as an abortifacient. There are a number of herbs that can be helpful in late pregnancy that could induce a miscarriage if used early on, so these usages don't seem all that contradictory.
Kava lactones are lipid soluble which would allow for a significant amount of them to pass into breast milk. Because kava can have definite effects on neurotransmitters, my personal opinion is that kava should not be consumed at all if nursing a child. Infant brain development is incredibly sensitive and changing the balance of what is naturally occurring could have detrimental, possibly devastating effects. Someone please correct me if I am wrong, but I don't think messing with the GABA receptors of an infant or very young child is safe.