I'm not sure if I understand the relation. Unless, of course, one isn't following with his arm but allowing an angle to be formed between hand and forearm. Now that I think of it, what exactly does "wrist motion" involve? 
The problems with this sort of question are really two:
1. Terminology and knowledge of anatomy: people use the word “wrist” and “wrist movement” to refer to all sorts of things, so that in order to even begin to discuss the subject one needs to write a lengthy post simply explaining the basic facts and providing a common nomenclature.
2. The best way to discuss any of this is through demonstration, not through writing. Writing just takes too long and is simply too inaccurate leading to all sorts of misleading and misguiding information. Two persons who share a certain knowledge will be able to talk about that knowledge and totally understand each other
even if their language is totally inaccurate and off the mark, since as they know the subject they will be able to make the necessary allowances and adjustments. However, if one of the persons does not have the knowledge of the other, or have a knowledge of a different kind, verbal discussion more often than not will degenerate in heated argument. So we come to the paradoxical situation where important information can only be meaningfully communicated through language to someone
who already has that knowledge, and therefore does not need to talk about it at all.
So.
First understand what is the wrist from an anatomical point of view, which is very different from the understanding most pianists have of it (and the common person in the street).
Look at these pictures:
https://www.ucgc.org/segments/wrist-injury/Most people would not call this the wrist, but would call it the hand, since we tend to regard the hand (minus the fingers) as solid block of flesh and bones, instead of a collection of 8 different bones.
You could argue (as xvimbi has done a number of times) that anatomically speaking the fingers extend well inside the “hand”- the metacarpal bones, but this is not what is commonly understood as the fingers. The common usage of the terms “hand”, “fingers” and “wrist”, is simply that the fingers are the phalanxes, the hand is the solid bit to which the fingers are attached (that is, anatomically speaking the carpals plus the metacarpals – so that in common usage the hand includes the wrist and sometimes the fingers too) and that attaches itself to the forearm, and finally the wrist is the “hinge” between the hand and the forearm.
So for a discussion of this subject to be meaningful we have to define (not that difficult actually) what exactly are we talking about.
Consider the wrist anatomically. If what you call the wrist are the carpal bones, there is really next to no movement in them. They move together as a unity with the metacarpals. So talking about wrist movement is simply an inaccurate use of language. When people talk about wrist movement they are talking about something else altogether – as we shall see in a moment.
What about from the point of view of calling the hinge between the hand and forearm (more accurately between the wrist and forearm) the wrist (improperly)?
Again a hinge does not really move: it allows the parts attached to it to move. So again there is no “wrist movement”, what there is is hand and forearm movement.
So this discussion is not really about wrist movement but about this: Should the forearm and hand be allowed to move by hinging on each other, either horizontally or vertically, or should we brace the hinge so that hand and forearm work as a unit? The answer – as with everything else in this universe is: It depends.
What does it depend on? Now again we have to go into another direction if we truly wish to understand the subject and profit from this understanding. And what we have to understand is the idea of levers and fulcrum points, since ultimately that is what the joints of the arm (shoulder joint, elbow joint, wrist joint, finger joints) do: not only they act as hinges, as they function as
fulcrum points for the bones (which function as the levers). The whole thing is of course held together and dependent for its proper functioning on muscles, tendons, ligaments and sheaths.
Wrist injuries are the most prevalent ones (see reply # 15 above). They are mostly caused by “up and down” movements of the wrist, and in the case of piano playing, if one plays consistently with wrists dropped, carpal tunnel syndrome is assured. Side twisting (abduction and adduction of the hand in relation to the forearm) - usually from trying to reach the notes with the thumb is another big no-no movement.
So, if you technique is based on up-and down and sideways movements of the wrists, and if you think that you are being emotional and expressing deep feeling by an exaggerated “choreography” of the wrists, think again, since soon all you will be feeling is pain, and all you will be expressing is regret that you did not pay attention to these matters.
The movements that should make the basis of one’s technique are:
1. shifting of the arm to position the fingers (so that the fingers to not “reach for the keys).
2. Rotation of the forearm to lift the fingers (instead of the “lifting fingers high” so prized by Hanon and their supporters)
3. Back and forwards movements of the arm to negotiate the black keys and bringing the thumb and little finger in position. (instead of twisting the hand on the horizontal plane to accommodate the short fingers).
4. Forearm and hand moving as a single unit to transfer weight to the keys (that is, bracing the wrist – instead of letting the wrist collapse or lifting the wrist exaggeratedly).
99.9% of the passages in music can be negotiated by using these movements and their co-ordination. This will guarantee effortlessness in playing and absence of injuries.
These three movements and their co-ordinations:
1. lifting fingers high.
2. exaggerated up and down movement of the wrist.
3. exaggerated horizontal movement of the wrist (adduction and abduction)
should be used only on truly exceptional circumstances, when nothing else will do. And in fact I would not touch any piece that would require (for me) such movements for extended periods of time.
This all leads to the principle of joint alignment. Use the “good” movements and their co-ordinations and you will notice that your joints are aligned, while the “bad” movements always result in severe misalignment of the joints.
Of course this is just the tip of a monumental iceberg and whole volumes could be written about it. As of necessity much has been left out, and the only efficient way to learn about these things is to have the good fortune of meeting someone who truly knows about it and is prepared not only to demonstrate it to the student but also observe, and correct the student – it is surprisingly easy to understand the concept and yet to be unable to actually do it. This really requires hands on approach.
Best wishes,
Bernhard.
Topic: how much wrist action?