Persi Diaconis gave a talk about probabilities of carries at MIT about half a year ago. Unfortunately I was so busy trying to spot symmetric functions in the talk that most of it drifted past my mind, but here are a couple references:

Persi Diaconis and Jason Fulman, Carries, Shuffling and An Amazing Matrix, arXiv:0806.3583

Noga Alon, Minimizing the number of carries in addition, arXiv:1209.1131

This all seems to be more about the idea of elementary school addition rather than actual digits of numbers, but the OP seems to not be excluding this.