I have never even seen one of these crossover stocks on a shotgun, much less shot one. However, it would seem physically impossible for one that is configured like that shown by Greener in the bottom drawing, exhibiting extreme cast, not to recoil offensibly into the shooters cheek, and also slide the butt outward across shooter's shoulder. The Parker pictured has this type stock. The Browning looks to be more like the top Greener drawing.
I have seen men who mount a shotgun improperly, with a body stance that faces to the right of the axis of the bore (rt. handed shooter), have horrible bruising from recoil upon shooting high volume, even with a 20 ga. gas gun and 7/8 oz. loads. This is from the gun sliding on the shoulder in recoil, and has the same effect as would the stock in the bottom drawing. One person was still bleeding through his shirt, because of his poor mount and stance, three days after returning from Cordoba.
Perhaps Bill will comment, he shoots crossovers.
SRH