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You may have a fungus that is attacking the plants. This would stunt the growth and cause leaf loss. Try treating the plants with a fungicide to clear this up. As a precaution, you may also want treat for pests with a pesticide as this may also cause issues on the plants.

There are many, many reasons for this to happen, from disease to cultural to environmental issues. It would be best if you had a local professional come and look at the damage as a visual examination is needed to really diagnose most tree problems.

Beech trees hold onto their foliage longer than do most deciduous trees. In fact, they don't always lose all their leaves in winter, making them fairly easy to identify. The dead, rusty-brown leaves can actually be quite attractive against a white, wintery background.

Normally these trees can be difficult to propagate via cuttings. That being said, hardwood cuttings are typically the type used for most trees. These articles should help: https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm

In looking through the forums, I see that these trees are very beautiful spring and early summer, then the pink leaves fade. Sometimes they get kind of bronzy, sometimes they get all brown around the edges, sometimes they turn all brown and dried. Seems to be the nature of the tree. You could try increasing the moisture level, maybe add more mulch, trying to keep it more moist. Perhaps more organics like peat or compost worked into the soil would help.

Beech tree buds are usually brownish in color. Since there is green under the bark the best thing I can do is prescribe some patience. Trees have their own timeline for breaking bud and they do so only when they know "true spring" has arrived. The following article will help explain:
http://northernwoodlands.org/outside_story/article/how-do-trees-know-when-to-leaf-out-in-the-spring

I wish things would green up faster too - I have had it with winter!

For more information on growing beech trees, please visit the following link:
https://www.gardeningknowhow.com/ornamental/trees/beech/beech-trees-in-landscapes.htm