Depends on how you handle the input.
In reality. No. There probably wouldn’t be any delay.
To do it properly the dog root bone would be animated inside the trainer, and the trainer would then manage everything.
The dog skeleton will just be told what local animations to play but it will also be linked to the root bone of the trainer.
In non optimal situations an interface call takes maybe between .0001 and .01 seconds to fire.
Usually unless the delay accumulates overtime somehow (repeated key presses) you’ll never really see anything go out of sync…
and there’s ways you can force a reset, like starting over the main character animation sequence at the same time the interface call is called.
Overall, you are way better off with separate skeletons/meshes when it comes to development and future proofing.
And stitching them together if you have the chops of animating them manually, since you’d have to jump through hoops to get motion from A to B.
Ps: I think I will try to put a rokoko suit on my dog at some point, but even if I do (and I leash him to a treadmill for science), I seriously doubt the quality would be as good as that of humans, So… wondering what you plan to do on the dog front…