[SideburnsOfDoom]

> The elites in Britain basically didn't seem to care (and perhaps encouraged) the importation of labor to drive down market prices of labor both for untrained as well as trained labor

And you have hit on a basic truth there. A fair part of the vote to leave was because of economic hardship. A fair part of that is because of the perception of "people coming over here and working for less". But every employee has an employer. Government and business can do a lot to set industrial policy, starting with funding for education (for doctors, nurses etc), and minimum wage levels (for field & factory workers).

Emphasis on "can". They didn't.