OK, you physicists confused this engineer. How are you shutting off the charged particle source? Is someone gonna turn off the sun? That is where all the charged particles originate from.
This idea is science fiction, and poor science fiction in my opinion.
It's an engineering problem. Sure there is physics involved, plasma fields and Jacobians and all that stuff. But the source will keep pumping charged particles forever for all practical purposes. However, the source is sporadic. Sometimes there is a lot, sometimes there isn't. Once the particles are trapped in the Van Allen Belt as it is today, they stay for a long time. If they were drained quickly, then the level would drop and stay low until the next solar outburst. So, overall, over time, the density of charged particles will be reduced. It's still a sci-fi idea, but a space tether ought to be launched just to see what it does. Who knows, it might work or it might just scare people into learning a little about the universe besides what is on cable tonight.