I've found that, for me, whether the webbing slips depends a whole lot on the stiffness the cord/rope tied to the rings, the knot used to tie to the rings, the stiffness of the webbing and even the direction I thread the webbing through the rings.
If the rope is pretty stiff, a simple larks head or a full turn and 2 half hitches will tend to make the webbing slip if you thread the buckle in the wrong direction. If you look at the rope and rings from the side, you can see that the stiff cord pulls more on one ring than the other with the larks head. The full turn with 2 half hitches does the same. If the cord is very flexible and the webbing is fairly flexible, the webbing will pull the rings even and help alleviate this problem. If the webbing is fairly stiff, then the webbing cannot pull the rings even. If the webbing is then threaded in the opposite direction from the cord pull, the webbing holds. If the webbing is threaded to enhance the uneven pull from the cord, then the webbing will slip.
3 solutions that I have found to work for me:
1. make sure that the cord and webbing are both fairly flexible and the problem doesn't seem to appear. The webbing can pull the rings even and the cord is flexible enough that it will conform to the pull by the webbing.
2. if the cord is fairly stiff, then do not use any knot like a larks head or full turn and 2 half hitches. Any knot in which the force is transmitted mostly by one thread of the knot can make the webbing slip in this case. You need a knot that will pull both rings equally. I have found that wrapping the cord around the rings at least twice and preferably at least 3 times and tying back with a bowline will work. This arrangement pulls both rings equally no matter if the cord is stiff or flexible.
3. If the webbing is fairly to very stiff, then you will definitely need to go with the wrap and bowline solution since the webbing pulls unequally on the rings and even a little unequal pull by the cord can cause slippage if the unequal forces from the webbing and cord line up.
Hope this helps in understanding why the webbing may slip sometimes and not other times.
When you say "even" rings, is it ok if they are slightly uneven? That's how mine are with my Hennessy Spectra attached to the rings. No problems with slipping. As I tie the bowline it is a little hard to get both sides of the spectra to be exactly the same tension.