Can I use a smaller diameter hot glue stick on a bigger diameter hot glue gun?
Updated: I've tried it. I don't recommend it. 

I needed my big glue gun this afternoon and had some small sticks lying around so I gave it a go. Melted glue came out where the stick goes in, then hardened. It was a job getting it out - several minutes of poking around with knives, wire cutters and screwdrivers, and narrowly avoiding getting burnt. I got more glue out of the wrong end than the right end. 

 I had previously used black glue; that's what you can see just above the "80 Watts" label. The thin stuff I was testing with is clear. As I took the picture the black stuck itself to both the clear and the feed mechanism. 

You will (I previously said "may") find that melted glue comes out where you push the stick in, so you will need a plan for cleaning up before it solidifies, or have a pusher handy (such as wooden dowel - a glue stick of the diameter the gun is used for didn't help).

There is one possibility of damage or more likely scorched glue: some big-stick glue guns (like mine) have two temperature settings. The hotter may be too much for small sticks (which I've only ever bought/seen as cool-melt).  So be sure to use the right settings, especially as you may have glue hanging around in there if it's not feeding very well. 

I'd always buy the right size, but would use only use up thin sticks in a bigger gun if I was desperate.