Rooting cuttings in moss has become my favorite method of choice. I have not needed to wash the fig cuttings before starting the rooting process, but should that become necessary, Pete S. in the above comment has a good procedure. I have tried Root Riot, wrapping in damp paper towels and placing in Zip-Lock bags, planting directly in potting compound, Perlite and sand etc. with limited success. With all of the methods except the moss, I had a difficult time maintaining the moisture level to my satisfaction or more importantly, the cutting's satisfaction. I have had a very high success rate with the moss. I also have no issues with mold or gnats because I'm not using soil.
All you are trying to do with the moss is to fool the cutting into thinking they are in a high humidity area. With the moss holding the water, the cuttings stay damp not soaking wet.
Here is my method:
First, I soak some long fibered sphagnum moss in a bucket of warm water to hydrate it. Let it soak for at least 30-45 minutes before trying to use it. It soaks up water pretty fast but it takes a while to get it soft, fluffy and fully hydrated. I then take a handful of it and squeeze out most all of the liquid to the point it is just damp. Using a plastic Wal-Mart shoe box (with snap on lid) as my container, I begin to pull the moss back apart and sprinkle it into the bottom of the box being careful not to pack it in too tight. You want it to be sort of loose and aerated. I lay this in until I get around 2 inches or so on the bottom.
Next, I identify the base of the cutting and dip it in some rooting hormone to cover the lower two nodes. Usually about an inch but no more than two inches of the cutting. I prefer the Clonex gel because it covers very well and it will soak into the cutting in a short amount of time. I then place the cutting on the moss making sure it is in good contact with the moss and repeat until I have 8 to 10 cuttings in the box. You can put in more or less cuttings as you want, it is just that this works well for me. I will usually mist the cuttings with warm water to give them a head start. I then cover them with some more moss using the same method I used to cover the bottom. You know, squeezing out the water and sprinkling it in to cover the fig twigs. I like 2-3 inches of moss over the cuttings. Another spritz of water and snap the lid on tight.
Now this is what I think makes my success rate go up. I place the shoe box on a heat mat made for seed germination and leave it on this mat for the entire time it is rooting. It never gets too warm for the cuttings and it really gets the box to sweating. The temperature gage in the box reads between 70 and 75 degrees. After about a week, I open it to check the moisture content and to air it out. If I think it is drying out too fast, I spritz it again, otherwise I just replace the cover. I also peak at the cuttings to check their progress. When checking on the rooting progress, be careful incase it is growing roots so you don't break them off.
I have seen the nodes start to swell and form those white bumps you are looking for that will become roots later after only a week to ten days on the mats. After about three weeks on some of the cuttings, there will be roots growing in the area where you applied the hormone, with the remaining cuttings rooting over the next few weeks.
Be sure to properly label the twigs and date them too!