I'm not really familiar with the Allegorithmic tools, but I can certainly explain how the material compositing works for The Order. Our compositing process is primarily offline: we have a custom asset processing system that produces runtime assets, and one of the processors is responsible for generating the final composite material textures. The tools expose a compositing stack that's similar to layers in Photoshop: the artists pick a material for each layer in the stack, and each layer is blended with the layer below it. Each layer specifies a material asset ID, a blend mask, and several other parameters that can be used to customize how exactly the layers are composited (for instance, using multiply blending for albedo maps). The compositing itself is done in a pixel shader, but again this is all an offline process. At runtime we just end up with a set of maps containing the result of blending together all of the materials in the stack, so it's ready to be sampled and used for shading. This is nice for runtime performance, since you already did all of the heavy lifting during the build process.
The downside of offline compositing is that you're ultimately limited by the final output resolution of your composite texture, so that has to be chosen carefully. To help mitigate that problem we also support up to 4 levels of runtime layer blending, which is mostly used by static geometry to add some variation to tiled textures. So for instance you might have a wall with a brick texture tiled over it 10 times horizontally, which would obviously look tiled if you only had that layer. With runtime blending you can add some moss or some exposed mortar to break up the pattern without having to offline composite a texture that's 10x the size.
With UE4 all of the layers are composited at runtime. So the pixel shader iterates through all layers, determines the blend amount, and if necessary samples textures from that layer so that it can blend the parameters with the previous layer. If you do it this way you avoid needing complex build processes to generate your maps, and you also can decouple the texture resolution of your layers. But on the other hand, it may get expensive to blend lots of layers.