What follows is my analysis of the cause and cure. But this in one of those issues that draws endless debate, so expect others to "correct" me. If you're less interested in theory and debate, scroll below the line where I tell you the fix. When you've read my fix, and the others pick one or more and solve the problem.

Cause.

Have you ever pushed a box or piece of furniture across the floor? If so, you've probably experienced the same issue on a grand scale. As you push (especially if you push from high) the box can hit a sticky spot on the floor and will begin to tip lifting the rear end. So you back off and let it settle back then start again. The same thing is happening to your brakes, but it's happening rapidly in a harmonic kind of action.

The shoe has two forces on it, the push against the rim from the brake arm, and the push forward by the rim. If friction is a bit high (good) the show moves forward, twisting the arm and digging the back end into the rim. That pushes the arm outward as it twists, and when enough pressure is built up, the rear corner lets go and the shoe is pressed flat again to start the process fresh.

It's a question of balance between the forces pressing the shoe down and forward, combined with the rigidity in the arms and pivots. You don't get the shuddering when braking hard, because the brake arm is forcing the shoe down flat, and it' cannot twist.

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The cure.

One sure, but stupid cure is to lower the friction between the pad and rim, but I'd venture you agree that it's not smart.

So, the common method is to toe the shoes in so as they push forward they move toward being flat, but resist the rear corner from digging in. This works 90% of the time. But if the arm isn't rigid enough, you need too much toe-in for this to work, and the brakes become very spongy, or may not open enough.

My cure, done after toeing in about 1/16" or so, has two parts.

First check for free play/rigidity of the brake arms on the post. If there is play, you want to eliminate it or dampen it somewhat. I've had success using a stiff grease on the bushing, and if the fit is too sloppy, sliding a piece or two of dental floss between the bushing and post to stabilize it.

Next (or maybe before taking the brake apart) use a coarse file, or course sandpaper on a paint stirrer to remove the rear corner so there's nothing to dig in and start the lift/twist. I file at a shallow angle, to about 2mm or so depth and 10mm toward the front, as if I were making a skip tip that will climb up and slide over any sticky spots.
That works 99% of the time.

Also, use a strong cleaner to degrease the rim which can accumulate a film of oil picked up from a wet road. Sine the film isn't even, it causes the problem, even on a brake that never had the issue (mine squeals for about 1/2 hours the day after riding in the rain, then fixes itself).

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So there you have it, a few things to try in any order until the problem is solved.

BTW- The problem is always worse on front brakes mounted in front of the fork. That's because the twisting action spreads the unsupported tips of the canti studs. Years ago they used to market "brake boosters" that helped resist that twisting, but they disappeared as disc brakes became dominant. Rear brakes can also squeal, but the twisting is a bit self limiting because the stud is pointing back, and cannot be spread at the front.

Note, I didn't mention the headset, but an absolutely tight headset with Zero play is critical to a fork staying stable during brking.