Soooo... As nobody responded with any advice or experience (presumably because nobody else has tried it) I thought I would just go ahead and try it in the washing machine. I put the dye packet and also the dye intensifier that came with it into the washing machine and ran a 60 degree synthetics wash. I paused the machine lots of times to give the covers plenty of time to take up the dye.
I just threw in all the old used white covers I have, but also two new small Rikki Wraps, which I was quite worried about. It wouldn't have been the end of the world if the others had been ruined, but I didn't really want to ruin the new Rikki wraps.
Of course it was the Rikki wraps that gave me the most trouble. I noticed after a while that they were splotchy with lots of little white gaps that hadn't taken up the dye. Then it occurred to me that I probably should have prewashed them
So I took them out and washed them by hand and then put them back in. This improved things but they have certainly dyed the most unevenly.
Now for the results! I have to say it turned out better than I expected, but I would have been happier if the outsides had dyed as well as the insides.
Anyway, here are the photos:
First, the inners -
And the outers -
On the top (l-r): Tots Wrap, Little Lamb wrap, Motherease Rikki wrap
On the bottomw (l-r): One Life wrap, Imse Vimse wrap, Motherease Airflow wrap
I did three of the tots wraps (one turned out quite splotchy on the outside, maybe it had something spilled on it once that left a residue), three LL wraps, two Rikki wraps, also a lollipop wraps (which as far as I can tell is nearly identical to the One Life wrap). But I thought one photo of each is enough
I have also taken individual photos if anyone wants to see anything in more detail.