Much of the cow diet is actually corn and soy, especially for US beef production cattle. There are emissions from the fertilizer inputs to corn and soy as well as emissions from the transportation of corn and soy from field to cow.
Ah that makes a lot of sense. Here in NZ most of our cattle are grass fed with very little feed transportation. Some farmers do use feed pads but when they do they are being fooled by the big dairy companies putting pressure on the farmers to get the milk solid production up. The tends to mean the farms revenue goes up but the costs go way up so they increase work and decrease profit. So not many of them keep up the practice. I'm guessing our climate allows us to farm differently, as we have few droughts and warm winters.
Highly depends on the specifics of a particular farm’s operations. Not a great generalization to make about dairy without supporting facts (that have got to be out there, so cite them).