There are some ideas that could be repurposed from asteroid mining concepts. Many asteroids are more like pile of gravel loosely bound by gravity than solid rocks, and either way, trying to extract anything risk launching lots of tiny pebbles in a random directions. One idea I saw floating around NASA website was to wrap such an asteroid in a net or sheet of material, to keep it from shaking itself apart. I could imagine similar lightweight net being used to wrap a satellite.
Once wrapped, you can do two things with it. You can try to deorbit captured trash - for instance, with a light net and a solar sail attached (which would be used to increase atmospheric drag). Such trash would not only deorbit faster, but a sail would make it easier to track. Nets and sails would have to be light and disposable, they could be shipped to orbit in bulk, or (later) manufactured in space.
Alternatively, you can try to recycle the trash. If you have in-orbit refueling using propellant sourced from space, you probably already have - or are about to have - some facility in orbit that could reprocess parts of dead satellites. So you can move the trash to some designated recycling orbit.
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Now that I think of it, you could probably make a recycling collector work without all the infrastructure in space. In orbit, you have time, you can take things slow. I can imagine a collector chasing light trash using ion thrusters, or whatever is most promising in high-Isp department these days (how small can a VASIMR get anyway?), grabbing it and parking at the recycling orbit. Might take weeks per piece of trash, but if you launch a bunch of such collectors and feed them trajectories from the ground, they'll eventually clean the orbit out of light trash. It might require some actual developments in the beamed power department, and perhaps launching a power satellite to microwave some electricity to the collectors, but think it would work in (sufficiently modded) KSP :). Anyway, the biggest problem isn't mission design, but figuring out who will actually pay for this.
Not sure what to do about the most dangerous space trash - the tiny bits that can't be reliably tracked. Track them better and vaporize with lasers?