I looked for a source for this, and found the opposite:

> Newborn oxytocin levels were higher in the umbilical artery vs. umbilical vein, and both were higher than maternal plasma levels, implying substantial fetal oxytocin production in labour. Newborn oxytocin levels were not further elevated following maternal intrapartum synthetic oxytocin, suggesting that synthetic oxytocin at clinical doses does not cross from mother to fetus.

In other words, the fetus makes its own oxytocin during labor. It does not come from the mother.

[1] https://link.springer.com/article/10.1186/s12884-022-05221-w