Conversion of non-Jewish babies to Judaism

I’ve been seeing a lot of conflicting (and often very vague) information on the web about the process of conversion of non-Jewish babies (i.e. born to non-Jewish mother) to Judaism.

In a scenario where a Jewish man marries a non-Jewish woman and has a child, what are the options for child conversion for each denomination?

I’ve heard that no Orthodox rabbi will convert a baby unless there is evidence that the child will be raised Jewish, kashrut, Shabbat, education etc. It’s not clear if the woman needs to convert as well.

As always, the initial Halachic source seems fairly straightforward and lenient, but in practice there seems to be compounded stringencies.

Thanks in advance!