Every character is different,and they all have their own unique backstory and history which will affect their decisions, acting choices, desires, and so forth, but if we're talking about body mechanics, there are many things that are simply right or wrong, independent of that specific character's traits or motivations.
Timing is something that will be affected greatly by motivation - a character going up some stairs will certainly climb the stairs at a much different speed if a Tiger is chasing them than if the authorities are waiting at the top of the stairs to arrest him - however, the actual body mechanics would be pretty much the same.
When we talk about body mechanics, we're basically talking about how the body works together. What moves what, and why. These things are based on our anatomy and the physical rules of our universe (gravity, physics, etc), not on personality traits or situations.
The man running from the Tiger and the man about to be arrested both have the same basic body mechanics. Each must put his right foot onto a stair, anticipate down and shift his weight to the right, rotating his chest in all three axis subtly as his left arm starts to move back, all of which allow him to start to raise his left foot to the stairs.
Tiger-Victim will be moving through those motions much faster than the Accused Criminal, but essentially they'll be doing the same thing with subtle differences. Tiger-Victim, because of his velocity, will have his weight much more forward throughout his flight up the stairs, and will have less up and down to his steps, and less obvious weight shifts, probably. Accused Criminal's slow and fearful ascent will be characterized by much more deliberate and obvious weight shifts, body weight that is strictly above the feet, and a much more upright posture.
But these are small differences within the basic body mechanics required to climb a flight of stairs. They're doing the same thing, just at different speeds, and the different speeds demand slight adjustments to the poses, but the basic foundation of what has to happen is still there for each of them. In some ways, Tiger-Victim is simply exaggerating some of the mechanics while toning down others - but the mechanics remain.
As for the general idea of initiating movement, it's usually a good general rule to think of the hips as the engine of any internally generated large body movements for your characters. It tends to be the first thing we move, probably because we can't move anywhere until we unbalance ourselves, and the hips are the center of our mass and weight. In order to move (take any steps), we *must* become unbalanced, and the only way to do that is to move your butt!
Of course, externally motivated movements are a whole other matter, as a frying pan to the head is going to cause your character's head to move before the rest of his body!
So, in closing, I guess I'd say to move your butt and avoid frying pans.
Good luck and have fun!