It is one thing to be against poverty; it is something else to be against inequality. One could end poverty and still have plenty of inequality; one could end inequality by impoverishing everyone.
What’s as important, focusing on inequality means making envy an engine of public policy, an odd status to give something that has the distinction of being one of the Seven Deadly Sins and prohibited by the Ten Commandments. In “Screwtape Proposes a Toast,” C. S. Lewis warned against encouraging people to embrace the closely related and obviously false notion that, for one and all, “I’m as good as you are.”
I would add that the notion of “white privilege” has the same lineage. The Left not only dislikes any standard with a “disparate impact,” it wants to put down those who have whatever it is that separates them from others. If someone built something, it wants to believe that “you didn’t build that”; if someone has something that others do not, it must be because of some illegitimate privilege.
And to the extent that there is less social mobility now, it is likely at least in part because the less wealthy lack the social capital they used to have. As Charles Murray has documented — and just by looking at non-Hispanic whites, by the way — the have-lesses are now more likely to be illegitimate, irreligious, criminal, and lazy than they used to be; the haves, not so much.