The same thing happens to the third party that has always happened. Their issues get taken up by the main parties if they are worthy. That's what happened to Perot's main issue in 92. Both the Dems and GOP took note and added the "balance the budget" issue to their platforms. So, it worked out well for everybody, but the Reform party. We got a balanced budget and the Reform party is toast.
At the local level, we still have the same issues. Right now we have a fight over our local RTA trying to grab land for shuttle parking. Some factions would like to get rid of it altogether. I voted for a .25% county tax increase to fund it about 20 years ago. It was supposed to be used to help "old people" get back and forth to the store and doctor. Now all we have is overpaid drivers and bureaucrats with a bunch of empty buses driving around. At this point I'm on the side that says pull back that tax. Start over and let the old people use vouchers to ride in privately owned buses, vans or cabs. Not far different from the Federal level, is it?