Before Brexit, we could have lived and worked in any EU country, including the right to move from one EU country to another, while keeping the same rights.
A) the right to stay living in a EU27 country indefinitely (though may be conditions, just as there were before Brexit)
B) the right to move to an EU27 country before 31 Dec 2020, in which case A) applies
So say you live in Spain or move to Spain before 31 Dec 2020. You can keep living in Spain after 31 Dec 2020 under the same conditions as before. But you couldn't just up sticks to e.g. Germany.
Once the WA deadline has passed, UK citizens not already living in the EU will be treated the same by EU countries as other non-EU citizens are. We're not special.
TL/DR summary...
- Before Brexit: rights in all EU countries
- Before WA deadline: rights in one EU country, preserved past deadline
- After WA deadline: no rights
NOTE: that covers everything except Ireland/NI which is complex and too long to go into here.
So EU citizens have the right to move to the UK and/or stay living in the UK before the WA deadline. If they do, they can stay beyond the deadline.
What they can't do is:
1) live in the UK for a while until after the WA deadline
2) move to an EU country (either their own, or others)
3) move to the UK again
But to do so they'll have to fulfill the same visa conditions in future as people from outside the EU did before Brexit happened.