Your health-food store will probably have beeswax in blocks, pellets, or flakes. Failing that, they’ll have pure beeswax candles that you can melt (after removing the wick and wick holder, of course).
Here’s how:
Get a foil pie plan ('cos the beeswax will trash a regular pan), and put it over a pan of boiling water ala a double boiler.
Put the beeswax in and heat it 'til it melts, and keep an eye on it – I wouldn’t boil it.
Once the wax is melted, add your oil a few drops at a time and stir the liquid wax concoction with a take-out chopstick or some such disposable implement. You’ll have to move fairly quickly because it will start to harden as soon as you remove it from the heat.
Pour it into a foil-lined Altoids tin (that’s what I use, anyway – it’s wonderfully portable, and you can take the foil out once the block has cooled); or, if you’ve gone for the goopier version, a jar.
Let cool, and bingo. You’ve got beeswax gunk that doesn’t stain your flute. It’s a beautiful thing.

As for plugging the holes, the beeswax/oil mix will warm in your hands as you work it, so just grab a chunk, roll it around like silly putty, and plug it into a hole. From there you can use a dowel rod that’s slightly smaller than the flute’s bore to shape it to the bore from the inside, or you can even use your cleaning rod. It’ll just take a little more pushing around since cleaning rods are small.
Expect to have to work it a bit until you get the proper seal. But it’s really easy, and the beeswax is nicely workable while it’s warm. Plus you can always “rewarm and reshape” as needed until you get the right hole coverage.
Best of all, it smells good!
The old guys have been doing this for a long time, and I have to say, I think they’ve got it right.
Good luck!
cat.
P.S. Oh, finally … when it’s time to remove it, you can warm it with your fingers/hands, or even a hairdryer on low (not for too long, though!) and scrape it right out with a toothpick, your fingernail, whatever.