I was curious when Piper essentially abandons Diamond City to follow her new friend around, she doesn't make arrangements for Nat to relocate to her new home (whatever Settlement you choose).
While working on your Affinity with her, about halfway through, Piper mentions the reason she is avoiding Nat is she is concerned that she's being a bad "parental" influence, because Nat is becoming too much like her and she doesn't want that.
Which explains why Nat isn't able to be relocated initially.
I resolved the situation by telling Piper that you always have to be there for your family, no matter what.
At this point, I fully expected to be able to move Nat to Piper's new home (in my case, it's Sanctuary Hills).
But this doesn't appear possible.
Surely it's irresponsible to leave Nat unsupervised all the time?
I assume given her high unpopularity with Diamond City, finding a childminder isn't possible.
I reasoned that given only Nick and Ellie are on good terms with her, they might be looking after Nat when Piper isn't around.
I thought it would be better to have Nat at Sanctuary Hills, where the population aren't as hostile towards the Wright sisters.
Also, there is Codsworth, who has proven experience with being a childminder and he could look after her while Piper isn't there.