In regards to Brawling -- I would treat natural weapons as pretty much the same as any other body part or improvised weapon (i.e. you are not using it in a trained manner, simply rough and tumble).
As for the Natural Weapons themselves....
There are 2 main ways to handle this, I think (there may be others, but I am only going over the 2)
1) Individual Weapon Skills -- Much as with any unique or unusual weapon, the character should be allowed to develop an individual skill for using that weapon. This skill could and likely should include any appropriate Combat Moves....
2) As a specific Martial Art -- One of the nice things about Martial Arts is that they also allow for the use of Weapon Katas. So, if you have a Lizardman with claws and a tail, you could create a Martial Arts style that incorporates those natural weapons. And as such when using Kata Weapons, they can be used with any known Combat Moves that go along with that style....
So, for example, Lizard Fu contains the Combat Move, LegSweep. For Lizard Fu, I would allow this move to be done with the legs OR with the tail (should the tail kata have been purchased).
What I would not do, is to give the character free Combat Moves (other than the ones that are already free, like Basic Strike), nor should you give them free ranks in any skills either. Having a weapon (or two) that cannot be taken away is a huge benefit already, they will need to train with it to be effective, and training basically is the spending of Character Points...
If you want to create some special Combat Moves for those natural weapons, fantastic! However, again, do not make them free... The Basic Strike and other Basic Combat Moves are there SPECIFICALLY to handle the most fundamental types of attacks, regardless of the weapon (i.e. they apply equally to both manufactured and natural weapons).