O my goodness, Lindy-loo!!! What a mess! I thought it was bad when mine come in with cockleburs but those burs are a hundred times worse! Poor horses that get those! I can't imagine a foal with all those burs next to their skin. Yikes!
Besides cockleburs we also have buffalo thistle. I've been on a mission for the last 10 years we've lived here to eradicate both. I think water birds bring the cockleburs, however, as they are worst around the ponds. Even when we think they are all pulled up, more appear the next year. The big lake near us is very low right now. We have been out treasure hunting on it and the bed is nothing but a solid mass of cockleburrs. It must be the water birds that distribute them.
Probably the woodland animals are spreading the burdock. I don't think you could ever stop the seeding totally.
The buffalo thistle is much easier to control. Nothing eats them, to my knowledge, as they are stickery even when tiny. Their yellow bloom is a flag that says "here I am! Come and dig me up before my yellow flowers turn into seeds", so they are not hard to find. But you pretty well need gloves to get them out of manes/tails. And I think they have some sort of toxin in their tips, as they hurt. I always put thistles and cockleburs in trash bags or on the burn pile.