How to prevent router from grabbing board when doing face moulding?
Router bits have to be carefully checked against the type of cut being created. For the general cut there is a correct feed direction and there is an incorrect direction. For some bits used in a certain way there is no ideal cut direction.

The general principal is that the work piece should be fed into the router bit so that the incoming wood is pushing against the direction of bit rotation. If you feed the wood into the bit in the same direction as the rotation of the bit the cutting edges will most assuredly grab into the wood and try to pull it along. 

The picture below shows the concept for the proper direction doing an "outside" cut on the wood piece when looking down from the top of your router table. 

 

If you are trying to do some other type of cut then you should post some pictures so we can make additional suggestions.