Gut bacteria and milk

How Breast Milk Engineers a Baby’s Gut (and Gut Microbes) 

It seems quite a complicated story that is being sorted out - how mother's milk both encourages some of the right bacteria for the gut, and inhibits others.  In fact, it appears it passes on microbes within the milk itself, too:

And, of course, their study highlights yet another benefit to
breastfeeding. It’s unique in isolating the effect of a single (major)
ingredient of milk, but Kaetzel notes that breastfed infants also get a
wide spectrum of other helpful substances.

For example, it contains its own microbes. Lisa Funkhouser and Seth Bordenstein have speculated that the lymphatic system conveys bacteria from a mother’s guts into her mammary glands, where they can be taken up by suckling infants.

If pups that don’t get SIgA from their mothers have weird bacteria in
their lymph nodes, could they then pass on different microbes to their own offspring,
when the time comes for them to produce milk? “There could be some
really exciting transgenerational consequences from not ingesting sIgA
in mother’s milk,” says Hinde.