There is no "EU right to live and work" for third country citizens per se, immigration for third country nationals is a national responsibility and not an EU thing. The Withdrawal Agreement only secures residency right in the country of residence before the transition period was over, which does not include freedom of movement within the EU, moving to another EU country for work requires one to get a work visa within that country, the posted worker directive actually does not help here, it is meant solely for EEA/EU workers, for third country nationals only very short and time limited posted worker status is available, usually up to three months. If one has lived in an EU country for five years or longer he can apply for a long term resident permit in that country, and after he has acquired that can then enjoy limited freedom of movement in the EU, however, each individual member state has a very broad way to implement additional requirements for the long term residency status, so that again, is very much different in each individual member state, additionally third country long term residents in the EU have to apply for residency status in each new country they move to, which means having to jump through any additional immigration law hoops that country deems necessary. Another way of course is available for those entered the EU as a highly skilled worker under the EU blue card scheme, if they have stayed in the initial EU country for 18 months or longer they can move to any other EU country, however they do have to apply for an EU blue card there as well.
Of course there is always the possibility to marry an EU citizen, if that one moves country their spouse can legally follow them and work there as well.