In the novel Dragon's Egg, the human crew use one asteroid to swing other asteroids in place to counter the gravity of the neutron star. I understood that it was similar to a gravity sling shot, but I wasn't able to fully get how the crew were able to move the smaller asteroids in place using the big one. Can anyone explain that further?
I know nothing of this book, but I do know a little about N-body gravitational interactions. When N >= 3, you can do just about anything you want with a little propulsion, but it may take a very long time. This has been proposed by NASA (good approximate for hard SF) as a way of sending probes to the outer solar system: http://en.wikipedia.org/wiki/Interplanetary_Transport_Network
As far as the stability questions, it is known that you can put 7 or 8 equal-mass objects on a stable circular orbit around their center of mass: http://adsabs.harvard.edu/abs/1988A%26A...205..309S
After introducing an external mass and its accompanying tidal forces, I imagine there is still some stability regime, which could be pretty close to the NS, especially if the "asteroids" have WD density, their constellation could be on the order of 1km, while the NS radius is probably closer to 15km.
When a small spacecraft slingshots around a large moon or planet, it can greatly increase its speed at the expense of very slightly decreasing the speed of the larger body. Given a method by which to move a larger body, you could very simply cause it to move smaller bodies by traveling near enough to attract them. More complicated would be moving it in such a way that the smaller body was given a specific increase in absolute velocity, which would be the "reverse slingshot".
Not an answer but to give context:
There are eight asteroids: 2 large and 6 smaller ones. All the asteroids have been collapsed by injecting their core with magnetic monopoles that makes them collapse to white dwarf density. The large ones were originally 250km in diameter and collapsed to 100m.
The larger asteroids are on highly elliptical orbits.