Ok I'll take a shot here...
Those are ech. and yes the leaves root and pup with ease if you do things right, there's tons of info online about it.
Just in general, the leaves are very easy to snap of, just the slightest touch from a downward and/or sideways motion, and they pop off. If you drop one, even if it lands roots down and the leaves themselves don't hit the ground it can literally break off all the mature leaves.
If the plants are plumped up from lots of water, it makes them even more brittle, if they get dry and wilty, they're less brittle and more flexible.
I've never personally seen overwatering make them only drop their leaves, not healthy looking leaves like yours, they might rot off at ground level, they do that a lot, and that rot can push up the stem, then the leaves might drop, usually not, the rot creeps into them or they just hang tough.
I don't think any of that really applies to what's happening here.
This is just my own experience talking, but I've had a lot of ech. and similar plants over the years.
I think what's happening is one of 2 things, or the combination of the 2.
These guys are surprising sensitive to sunburn, and it can and does burn them up bad! Once acclimated they can handle plenty!
When they go from shade to sun you really have to inch them out into the sun.
The brown(dead) portions of your plants kind of look like sun burn, but it's only in a few odd spots, basically every time mine have been burned, it's very uniform, because the whole is exposed, the whole plant gets burned, one day can be enough to do the job and totally melt the plant.
That pretty much rules that out.
I the think the real culprit is water.
These plant have a glaucous coating, white powder or dust on the surface of the plants.
The powder on ech. is hydrophobic, and resists water, like a lotus leaf or many other plants... Or water off a ducks back!
When they get water on the leaves, it beads up, and runs off, but sometimes a little head of water gets trapped on the plant, the trouble spot is where the leaves meet the stem, especially towards to top of the plant, there's a lot of small leaves packed in a small area, but still enough air space to let tiny amounts of water in... Especially it rains really hard, or blows hard while raining... Or if you hit from the side with hose...
If the plants are tucked back in a spot that's sheltered and blocks the wind, it doesn't help!
All those holly leaves make me think it might be sited that way...
The top plant in the pic is what makes me think this.
It's pretty tell tale damage...
If it were mine, I'd pull the whole thing out first thing... Make sure that the soil is appropriate, gritty and fast draining, cactus mix basically, and the whole thing in a clay pot that was about as big around as the 3 plants are together.
That's assuming the roots look good and it's not rotten, but I don't think it will be...
After that put them in the brightest shade you can find and inch them out into the sun.
You don't want to put any water on them for atleast 2 weeks, probably longer.