Question: I've got a fortress that's grown up past its adolescent phase and is starting to mature. Naturally, I've had a number of encounters with goblins and werewolves. Since my primary defense is a hallway full of cage traps, I now have several dozen goblins chilling out in cages in my animal stockpile room. Although ostensibly I could keep them there forever, I'd much rather do something productive with them like establish a petting zoo, hold war-crimes trials, or start a goblin aquatics program. The problem is that every time I try to move the goblins and/or their cages from their current place in the stockpile, the goblin breaks free and runs about my fortress causing all kinds of havoc.

So: how do you successfully move a caged goblin? Either to another animal stockpile, a built cage, or to release them safely into a sealed room?

Answer: 
To safely release the goblin where you want it.

Build the cage, using `b`uild, ca`j`e, e`x`pand (to show all cages and their contents), and build the cage where you want it.

You can then connect a lever to the cage (`q`uery the lever, `a`dd a task to connect to the cage, and use that to release the caged creature (goblin or otherwise) by pulling the lever (add a task to `P`ull).

You can also use the take from stockpile command to move them to another stockpile.
You can dig out a room with a hole in the ceiling, designate that hole as a Pit zone, and assign the goblins to that pit. Your dwarves will throw them into that hole. I've found that thieves have a pretty good chance of escaping, but the others won't.

I have it set up so they fall 10 Z-levels into a small room, where I can decide to either send in my military for a little practice, or pull a lever and smash them with a bridge.

You can, supposedly, designate (d-b-d) the cages for discarding, claim (d-b-c) the same zone, then manually (k) undiscard the cages themselves to order the dwarves to take the goblins stuff and discard it into a garbage zone. I've had trouble with this, but that may be because my goblin stockpiles are in a very out-of-the-way portion of my fortress and I have many other discard jobs running.