In the Hillary Clinton case, it's not because she changed her opinion - it's because she
only changed her opinion once it was clear she'd face more backlash for holding it than abandoning it (indeed, the opposite appears to be the only reason she'd ever held her original stance in the first place). This indicates that, on future civil-rights issues to come that will have a similar public adoption curve (ie. transgender acceptance), she'll take the same strategy of appeasing the opposition until she can no longer get away with it - and, by turn, if the political winds blow the other way and the makeup of her constituency regresses toward a majority of bigots, she'd be perfectly willing to strip those enfranchised by her previously-enacted positions, so as to be continuously pleasing the majority.
Voters refusing to fall in for Hillary Clinton are holding out for a candidate like Bernie Sanders, who has made it clear that he will follow his moral compass, even when the people around him have been whipped into a hateful frenzy. Considering the history-repeating-itself rise of nationalism and racism in the face of economic hardship the world is currently barreling toward, this is the only kind of leader that Sanders supports feel could possibly help to right the ship.