@Aron there is no bouncing off the planet. This is the transfer of momentum. Look at the possible 2016-2020 jupiter flyby and return mission linked in the example query. It goes outbound with 6.28 km/s velocity and returns with 20.13 km/s velocity. The gravity assist could then donate 14 km/s velocity (on a probe, that's not much momentum - on an asteroid, its a bit more) back to Earth and head back out to Jupiter again.

@ckersch you don't. You send it out once. You slingshot it around Jupiter. It picks up some momentum. It slingshots around Earth and loses some momentum. You adjust the trajectory so that it will head back out to Jupiter (or Saturn or further out if the orbits aren't in alignment), and then repeat the process of slingshot and return and slingshot and outbound. It is transferring momentum from one body to another.

You supply the energy to adjust the orbit of a minor celestial body. You are transferring the energy for moving the major celestial body from another one. Momentum is conserved. You are not spending millions of kg m^2/sec to move the planet but rather picking up a a small amount in a gravitational assist from Jupiter and then donating it to the Earth. And then doing it again. As part of this process, Jupiter will slowly orbit closer to the sun, though it has lots of angular momentum to spare.

@JDługosz it doesn't take significant amounts of energy (when compared to the overall expenditures) to grab say, Ceres, Vesta and Pallas and put them on the proper flyby orbit for Earth / Jupiter. The less massive the body transferring the momentum it actually gets easier (less delta V needed to do adjustments) - it just takes longer as you don't transfer as much with each pass. But with sufficient time you can transfer momentum from the gas giants to an inner planet. io9.gizmodo.com/5923828/… suggests 1M passes needed.