Bicycle Mechanics - Avid Shorty 4 - cleaning pivot point

Ok, so after yesterday's 'cross race, I needed to clean all the mud and grit from my brakes. So I disassembled them as far as I could by removing the brake from the post and the pads from the brake. Cleaned that mess up, so far so good.

But, there's still grit inside the pivoting point of the actual brake itself. Not sure of the technical part name, but there are two pieces of hardware that rotate inside the brake pivot point. One is the bracket that holds the adjustment screw, and the other holds the spring.

I'd like to be able to take those hardware pieces apart to clean and re-grease. It looks like maybe there's a retaining clip deep inside the pivot that I can't access. Possibly a special tool is needed OR the brake just wasn't designed to be maintained at that level.

Anyway, any thoughts or suggestions are appreciated.

if they are the type of brakes i would expect (you were not specific to the make or design of brake you have) it should be center pull brakes as most cyclocross bike use this type of brake they should only need a soaking in a mixture of soap water and a little kerosene and then if you have compressed air blow them dry

I posted the name of the brake in the title (Avid Shorty 4) but not in the post details - sorry. They are center pull canti's.
Thanks for the suggestion - sounds like a good idea.

If the blast of air doesn't clean out the grit then you'll find that you can strip out the bushings from the canti arms without all that much trouble. The facing washer on the front of the arm typically is a press fit onto the nose of the bushing. A light prying force will lift this facing washer off the front of the bushing. Then you can slip the whole arm pivot apart for cleaning and adding a wipe of grease before re-assembly. One hint. Study well how the return spring fits into the cup at the back of the assembly. Lift off the canti arm without disturbing the rear cup assembly. Study it well. You've got a total of 4 chances to learn how it all fits together.... :D

solvent on a Q-tip should work clean off grease from bushing, and the grit should go with it..

then re apply fresh grease.