Eurasian Pygmy Owl and Parrot Crossbill (from http://niaolei.org.cn/posts/10555)
I thought that photo might get your attention ! No not from my Deeside site but rather somwhere colder and much further East ! Apologies to the anonymous photographer (as no name was attached to it) but I have linked to the original website.
Crossbills are strange birds; they are very wary when coming down to the ground to drink, get grit or bones (for calcium) yet when they are feeding in trees you can stand only a few feet from them and they are oblivious to your presence (helpful for reading rings !).
Even with my 1000's of hours field time I have never witnessed Crossbill predation per se eg. an actual kill, though I have observed unsucessful attempts by Sparrowhawks on several occasions, and Merlins (twice), both aimed at Parrot Crossbill flocks. Clearly Pygmy Owls have some adaptation for catching Parrot Crossbills ! Predation by avian predators it would seem is a rare occurence.
In Scotland there are several species that can predate Crossbill nests (eggs and nestling predation): Red Squirrel (Greys don't tend to occur in Pine Crossbill habitats), Corvids, Pine Marten, Great Spotted Woodpecker (and Green Woodpecker at one of my sites) and Sparrowhawk ( I have witnessed the latter though the chicks successfully 'exploded' and evaded predation). However, this said, the severity of the weather, particularly wet, snow and wind combined with the availability of cones (and how 'open' the cones are) will have far more impact on productivity than all the predators combined.