GIMP :: Time-based Gradient Brush

I'm quite new to the GIMP, though fairly experienced with Photoshop. I have a rather unusual requirement, and I'm wondering if the GIMP can meet it.

I'd like to set up a gradient and then draw freehand trails with a brush that gets its colours from the gradient, sawtooth style. So far, so good; I know how to do that.

What's unusual, is that I'd actually like the colours to cycle through the gradient by time, not by distance. In other words, if I move the mouse slowly, the colour sequence will be compressed, and if I move it faster, it'll be stretched out. Is this possible?

I seem to recall that there's an animated brush feature... maybe that's the way to go? Can it animate through a sequence of colours? If you're wondering why I want this, it's because I want to emulate what it would look like if you looked at the Crab Pulsar (from, say, a mere few AU) via a small pocket mirror, and rocked the mirror rapidly by hand.

By the way, 2.7 doesn't seem to have the "Use colors from gradient" option for the brush. Had to revert to 2.6 to match the online tutorials I found... In fact, in 2.7 I couldn't find any way to paint with a gradient (except "random").