"Ane Marie Johansen... was living in Vejle 10 months prior to the birth of Cecilie."
Paul can address this - but for the moment let me say that your comment on this isn't quite correct. Ane Marie Johansen was an unmarried, pregnant woman in need of support. Who would be responsible for that? The answer is the community in which she lived, 10 months prior to the birth of her child. If she moved during her pregnancy, the new community would not have this responsibility; rather, her previous home parish was responsible.
To my mind, this is like the current parish saying to the previous parish, hey you guys, these shenanigans happened under your nose, not ours. Pay up.
For that reason, when Ane Marie's child Cecelie was born, and the pastor noted the birth in the church book, he also noted Ane Marie's place of residence ten months earlier. This is what Paul read and is referring to. Cecelie was born in Vejle; Ane Marie lived in Vejle ten months prior; therefore Vejle is responsible.
As for the poor in Denmark "not being allowed to leave their locations" that sounds like a confusion of some partial truths and maybe is referring to the days of serfdom in Europe? But not necessary to go into all that in order to understand Ane Marie's particular situation. She was free to move. She can take a job in another parish. She can travel elsewhere to live with relatives or friends. She can immigrate to another country. Whether the support of Vejle parish would follow her (financial support would be sent to her), that I don't know. Maybe Paul does
So, just like you and me, yes, she's got to work out the logistics for her life decisions. But she is certainly "allowed to leave."
Interactive, clickable Map:http://www.dis-danmark.dk/kort/kort.htm