Rotation constraints don't matter with Mixamo animations

I have a model of a robot, that is humanoid, but his bodyparts (arms and legs) are divided to several units, each rotates only in one axis. At the same time I need this robot animated. I chose some animations on Mixamo, uploaded my model there. It created autorig for my model, then I downloaded one animation with skin. Besides the fact that I need each unit rotate only in one particular axis, everything works fine. But once I try to apply rotation constraints, it doesn’t work anymore, because animator is applied to parent object of the model, not to the rig, which I try to constraint the model to. Can I constraint the model to the parts of its avatar? Or what if I download an animation without skin, can I constraint my model properly to the rig in that case?