Freedom of movement within the EU means that if you are a citizen of one EU country you can legally move to another either to work, or to study.
If you go to another country looking for work, you can, if you qualify for it in your home country, receive unemployment benefit for a limited time. After that you can apply for it, or social security on the same footing as the citizens of the country you are in.
If you are unable to find work, you can be required to move back to the country where you hold a passport. Likewise if you commit a crime, you can be deported, either before or after serving a prison sentence in the country where you were convicted.
Whether you can as a citizen of an EU country live permanently in another EU country, is, as far as I know, up to the country you move to to decide.
The rules are complex, determined by such factors as whether you are in work and paying taxes, married to a citizen of the country you reside in, over retirement age etc.
Little of any of this is relevant to Northern Ireland or to Britain and has nothing to do with the border between Northern Ireland and the republic.
Brexit was always going to affect the movement of goods between Britain and the EU. We could clearly see this in the EU and took steps to meet the eventuality. Why the British could not do the same is a complete mystery.