Some more numbers may help. 400 kilometers altitude will reduce air friction losses. The asteroids need to be changed from solar orbit to circular LEO = low Earth Orbit. This makes them moonlets instead of asteroids. The tether = ribbon dragged by the 20 ton moonlet will need to be at least 500 kilometers long as it will likely be S shaped instead of straight. If the larger moonlet is 200 tons, the less massive moonlet will swing around the 200 ton moonlet, perhaps once per eight minutes, depending partly on the separation. So the lower end of the ribbon will dip into the ocean at 8 minute intervals, while moving horizontally at perhaps 20 kilometers per hour. Is this something like what you are suggesting?
The tip travels 2512 kilometers each 8 minutes so the average speed of the ribbon tip that is farthest from the moonlets is 314 killometers per minute = 18840 kilometers per hour. This is likely too fast, so 9 minutes per dip in the ocean is likely better. Timing and station keeping will be quite critical. It will likely be nessessary to reel some ribbon into the small moonlet each time the moonlets pass over land. alternately a tip motor could increase the S over land and stretch the S nearer straight over ocean.  Neil

I believe publiusr had asteroids much larger than 200 tons in mind. In principle, it would make sense to use the momentum of asteroids flying by to loft large masses into orbit. (Un)fortunately, suitable asteroids come this close to Earth only very rarely, and when they do it is not considered a good thing.

We could put up an artificial moonlet that would move in a highly elliptic orbit, with apogee at the moon and perigee in LEO. The moonlet would be built from moonrock and then gradually decelerated at apogee by a few tenths of km/s until it reaches its working orbit. At perigee, it will then whip around the Earth at 10-11 km/s and can theoretically be used to pull up a lot of mass from the surface. As mass is launched this way, the moonlet's apogee will decrease until its orbit is circularized. The momentum and energy for launch comes from the moonrock being lowered into the Earth's gravity well. Each moonlet could launch around one fifth of its weight. The spent moonlets would accumulate in LEO, where they could be converted into habitat shielding. It is a very favorable exchange, but there are substantial engineering problems in the mechanics of coupling a large load at the surface to such a fast moving target. A really large rubber band or spring would be required.

Andreas

That is exactly what I was suggesting. In this manner, large excavating machines can be bodily lifted off Earth (Front End Loaders, Caterpillers, wetstage Domes with gas propellants for RATO systems.

The asteroid would be moved close to Earth and the Moonbase assembled with this mega heavy lift system. No more shrinking payloads.

A 1000 km rubber band should do nicely to reduce acceleration to a few g.

Andreas