Well, well, what have we got here...! yummies, yummies...!

:biggrin:
Before I saw the pics, and reading your OP I though to myself, how big is this fellows' backyard...??? chezzz it has to be of considerable size for him to need to mark a trail back to the spot...!

:wink:
What a nice little "jungle" you got there mate, I would be in paradise, in there...!:wink: shame that I'm so far away...!:frown:

Now, and from what I see and what you said, the wood is old, wet and full of moss/fungi that will influence what's going to happen in a near future with it.
Apart from being cut from the tree, those burls need to be cut properly to give you as much chance as possible to make it usable when the time is right. There are a few different things that you could do to them, some more complicated and expensive others more doable and at this stage/moment, if I was there or if they were mine, I would do the following;
*- Cut the burls that are pretty much all around the tree with about 3" of wood/trunk on either side of the burl.
*- Any burls that are to one side of the tree, do the same thing, cut them as is the burl was all around with the same cut distance from the burl.
*- Take them to a cement slab area and water pressure wash them to remove dirt, moss and surface fungi.
*- The logs that have burl on one side, slice the log in half or so, removing the non burled wood.
*- Put them under cover for 24 to 48 hours.
*- Now that most of the water used to clean them has gone, use so exterior acrylic paint (any colour) to seal the full burled logs (both end grains), and the half logs with the burl still attached, paint the ends of the half log and the exposed wood from the log ripping in half. In general all fresh cut exposed wood should be sealed (painted).
*- Store them in a dry and away from weather location, this will allow the water inside the wood to slowly dry off, and reducing dramatically the rotten process some more advanced areas of decomposition my have. The wood that is still solid, will lose the excess water and will preserve well...!:wink:
Any possible stabilization or wood hardening work will be better done and achieved when you are ready to use each individual piece as is a good chance they will all have different decomposition and other stages, including pieces that will probably be needing no repair at all.
At this time, various processes and methods can be used but at this time, I would worry about preserving what you've got from further damage and what I suggested to you are the exact steps I would be taken in the wood/burls was/were mine...!:wink:
Good luck.
Cheers
George