Re: Sidewalk repair
If the gap is wide enough, dig them out and install bricks between the segments. It will be a nicer look that is repairable should the segments continue to move. If the gaps are not wide enough, you could cut them wider for the installation of brick. I would be inclined to loose lay the brick over compacted sand rather than trying to mortar them in. If you don't like those ideas, you might be able to tighten up the gaps by moving the segments closer together with a prybar. You'll have to dig around the perimeter to free it up. With a little luck, you'll be able to move them all tightly together, leaving a big gap at the slab that needs replacement, incorporating the extra space into the new slab.
The segment that is breaking up sounds like you'd be better off replacing it than trying to repair it, you'll certainly get a better looking job that way.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!