You might have a point there gnu, especially when it comes to city bird but pigeons are among the fastest fliers in the bird world. They are able to elude the swiftest predators except for the goshawk who seems adapted to the pigeons-for-lunch program. Or is it the Pergrin falcon? I can never remember which. Both are incredibly fast birds.

The pigeon is ungainly in stride because of its huge breast muscles far to the front and legs far to the rear. The bobble wobble walk is a result of this. They ARE ground feeder and this type of head movements are a common thread with other bird such as those of the gallinaceous variety (chickens, pheasant, turkeys, etc.). It seems like they stretch their head forward to the point where the body will catch up and then they repeat the pattern. Ground feeders are natural prey because of their constant exposure to ground based predators. Their peripheral vision covers all that is behind them but not all to the front which resluts in their cocking the head to one side to locate food. Re-cocking the head helps them triangulate on the morsel. They make a snake-like strike and snap off the bit of grub or grain at-beak, so to speak. The same striking movement serves well when establishing the pecking order. My Dad called that bottom-rung bird on the farm the "Peck Bird". You don't want to be the peck bird!

Dove are also in this family (passerine, I believe) and they also exhibit the head bob and they too are very fast fliers. I hunted the same in my youth and that is where I learned about leading the moving target. Unless they were coming straight at you or away you would never hit them by putting the sights dead on them. In WWII the Navy looked for gunners that had open field bird hunting experience as it served them well when manning the triple A batteries.