Without climate change inequalities might well increase, but with it they will increase more. I’m also saying that the wealthy may not actually be better off, but they may be relatively better off compared to the less wealthy (e.g. absolute wealth might decrease but relative inequality might increase). For example poorer people may lose land to flooding and go from subsistence to poverty.

Edit: lots of articles about this online, but e.g. http://time.com/5575523/climate-change-inequality/ has some sobering statistics on the likely impact of climate change: e.g. "A 2015 study in the journal Nature projected that the average income in the poorest countries will decline 75% by 2100 compared to a world without warming".