Darn it, I had a boomerang contraption that would at least occasionally solve Four Balls, but lost it when the power went out. Here's the basic outline, in case somebody else would like to recreate it...
Three balls roll down into a basket, with a lid that's propped up with a water rod. The fourth ball is actually part of the basket - that's the only kind of wheel that can be used, the regular sized ones won't fit in the goal.
A car, made as lightweight as practical, pulls the basket up a ramp towards the right. The lid falls shut, forcing the loose balls into a horizontal row, right on top of the wheel. The car eventually falls off the right side, dragging the basket into the goal behind it.
Meanwhile, there's a big, heavy, intentionally inefficient, left-moving car on a platform in the top half of the building area. The right end of the platform is lifted up by a crank (located under the ramp used by the right-moving car), to get it out of the way of the basket. The car pushes down a bridge ahead of it, to get it over the rolling balls. The car eventually falls off the left side; it trails a really, really long water rope behind it, also attached to the basket. Tweaking is needed to make sure that it doesn't reach the end of its rope until after the basket has reached the goal. A tug-of-war ensues, which the left car eventually wins because it's heavier.
The leftward movement might have been better done by a winch than a falling car.