When the National Guard deploys to situations like riots or sat the Mexican border, what types of MOS's usually deploy?

Every National Guard unit that I was associated with did riot control and crowd control training, whether it was Infantry, Engineer, Armored or MP. The governors usually leave that decision up to the State Command Structure. In general, the line units like those I mentioned above will get the bulk of that kind of duty and the rest doing what they are supposed to do, support the line units.
Both Battalions I served in also supported the State Prison system. The one here was a minimum security, the one from Ohio was a maximum. The training for that was interesting, they used the prisoners as training aid. No firearms on the ground, just the flak jackets, helmets and batons, with the guards behind the formation directing the movement of the unit. The guards would isolate on a troublemaker and then we would move over to pick him up. A reasonably trained squad with batons and shields can move some uncooperative bodies. Then we would open ranks enough for the guards to snag their target and protect them as we withdrew from the yard.