does family sharing work if members are in different countries?

No.

Share App Store, iTunes Store, and Apple Books purchases with Family Sharing - http://support.apple.com/HT201085 - "To share purchased content, all family members must use the same iTunes Store country or region."

It would have been helpful to know about this rule when we were reading about Family Sharing before buying new hardware.

There was also no mention of it when we actually signed up for Family Sharing. Even when we invited our third family member there was no warning about this rule or even an error message — so we kept reissuing the invitation and not understanding why it was never possible for the third family member to join.

I think these omissions are a serious lapse in customer care.

Apple products feedback links - http://www.apple.com/feedback/