I think if today's liberal progressives dialled down the globalist creed a bit and allowed people in countries to feel good about being part of a nation instead of an amorphous global 'culture', some of the illiberal backlash would go away.  

For example, few deny that the EU's economic union is a pretty good thing.  But the EU is so hellbent on melding the disparate nations of the continent into one entity, interfering with national policies, relentlessly imposing a ridiculous array of restrictions and products and the rest, that it's not listening.  About 1/2 the British, always standing back a bit anyway, said 'no' to this. Other countries may follow suit. 

Now's the time for the EU to listen to what its people are saying. But they aren't. All they're doing is wringing their hands and calling the British names.  The American Democrats right now are doing the same kind of thing.  

Come on guys, pay attention, or you WILL lose the plot to some ghastly demagogue for good.