Animator Transition between layers ?

I have an environment object with a few different moving parts (something like an elevator), and it has a few different animations that affect separate bones. One animation that opens doors, another that moves a ramp, another that slides some panels, etc. They were created separate so that they can be combined and tweaked in Unity.

One of the situations in the game requires a few of those animations in a sequence, exactly one after another, with simple Exit Time Transitions.

Even though the animation clips have proper Mask set up in the Import Settings, when combined in 1 layer in the Animator, all animations affect all bones. Creating separate Layers in the Animator Controller works, but I loose all the transitions.

One very annoying solution is to have separate Layers and separate bool Parameters for each part of the object, and than add Events at the end of the animation Clips, and trigger the different animations from script. Surely must be an easier way to link some animations in a sequence !? It would be very counterproductive to do it this way as I require sequences of about 7-8 animations.

Another solution that might work is if there was a way to create new animation Clips by pasting together the existing ones.

Is there anything that I missed about how you are suppose to set up Avatar Masks, or Layers in the Animator, or a way to do transitions? Or even some completely different solution to easily link a bunch of animations ?