I was thinking those cart boosters would look more like those log pushers in the woodcutting mills in skyrim. Not boosting them off at 50 mph more like hooking onto the undercarriage and pushing up an incline. Maybe with enough speed to get it up a single ramp.
Minecarts are heavy after all, and if the weight of the minecart exceeds the weight of the water pushing on the wheel the cart is going nowhere. Any gearing to increase the speed is just going to increase the effective weight of the cart against the system. Hell it would probably need to be downshifted in reality to get a loaded cart going anywhere.
Well since power is being supplied from a spinning axle, and, since a moving cart will either be slowed down or sped up when going through a "booster" it has to be either a Hot Wheels type accelerator with two oppositely spinning wheels positioned perpendicularly and on either side of the cart, or them positioned parallel but some how able to grip a section of the cart (like a lattice extending on both sides). Since Dwarves don't have rubber or any material that can grab (grind stones maybe?) I'd imagine a lattice on both sides of the cart which spinning "Hot Wheels-esq" gears can grab and accelerate.
The method from "log pushers in the woodcutting mills in skyrim," which I believe is the same method to accelerate aircraft off an aircraft carrier, would stop the moving cart upon hitting the "hook" from behind or require extremely complicated mechanics to get it to extend and grab the cart as it passes...etc.
But yeah proper conservation of energy isn't really a consideration in Dwarven contraptions.
EDIT: So I guess the real question here is:
What items will be required in the construction of a "roller?"