The difference when you use them as verbs describing human behavior depends on the kind of animal they are used for.
Herding is used for herd animals. If humans are said to be herding onto a sidewalk, they are to be pictured moving relatively slowly and in a somewhat orderly fashion.
Flocking is commonly used for birds, so when used for humans, the verb "flocking" would imply more bird-like behavior. It would be less orderly, and more quickly than herding.
Swarming is used for insects. If humans are swarming, then they are moving very quickly, probably pushing to get in a better position. Usually it is used in the phrase "swarming around [something]." Probably because humans rarely exhibit swarm behavior unless there are many people frantically or chaotically trying to get to the same place.
This only applies to verbs. A school of fish doesn't school around, so you wouldn't say that humans are schooling, without it explicitly being a metaphor.