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Isn't this kind of like Peter's Chess with Cyclical Armies?
http://www.chessvariants.com/other.dir/cyclical-armies.html
And I like the idea of polypiece Ultima or Rococo.  How about Polypiece
Optima?  Or Nemoroth?

Chess with Cyclical Armies is a wonderful example of the traditional way of
doing this, and I wish that I had mentioned it in my article!

The difference is that Polypiece changes not only the Knight that moved,
but all the other Knights!

Why is this Knight different than all other Knights? It isn't! They *all*
become Fibnifs, and not just your Knights but the other player's Knights
too!

Polypiece Cyclical Chess is predefined. The difference in the play of the
game is immense, even though the polypiece idea is really just a small
change to the way such things have been done before.

'some were 'ahead' a few steps'

Is this different from what I called 'Phased Polypiece Chess'?

I guess it must be because you talk about every move changing the whole
board.

You need to explain it better, give it a nice name, and write it up as its
own page.

I can sort of guess what you have in mind, and it looks like a nice idea
but you'd have to be crazy to try to play it, don't you think?

Of course, 'crazy' for some values we're all that here I guess.

My suggestion was slightly different from Phased Polypiece Chess. Suppose
FIDE armies. Suppose a list like this: R -> N -> B -> Q -> NNZZ -> R that
applies to all pieces of both sides.

After white makes any movement with one such piece, all rooks on the board
become knights, all knights become bishops, all bishops become queens and
the queens become zebra+knightriders, just as an example.

A game like this would, of course, be overly complicated.

It has occurred to me that this is a highly cool way to make
limited-number-of-movements chess, just by using a list that <i>does not
loop</i>. The last element of the list would be an immobile piece. This
way, a player would watch his army be slowly reduced to immobility, kind
of piece after kind of piece. Such a game should better not have a King
and be played for stalemate.

'all knights become bishops, all bishops become queens'

Ah. I was thinking you meant something like all N become Fibnif, all B
become FA, all Q become RN and so on.

As far as tending to immobility goes, that's one way to make a game of
limited length. Too bad that contest is finished!

So, work out the details of your idea, give it a try or two, and then mail
it to [email protected]; okay?

In Polypiece chesses, something happens and some piece-types change their move definition. Fourriere made Pocket Polypiece after Betza's idea:
http://www.chessvariants.org/43.dir/pocket-polypiece43.html. Both articles tend to change one-piece type at a time upon moving any one unit of the type, but Fourriere's goes further in changing them both ways, back and forth in fixed one-to-one correspondence. This is the first direct comment here in 7 years to the day +/- 5.  Revenge of the King below is different that capture of Queen, or return of Queen, changes one type's, the King's, move definition. There is a lot of unexplored territory and always will be -- Jaguaribe's world change.
Among a dozen other designers elsewhere, I did some CVs at ''91.5 Trillion CVs'' where movement rules or capture rules mutate whilst certain conditions hold:
http://chessvariants.org/index/displaycomment.php?commentid=18386,
http://chessvariants.org/index/displaycomment.php?commentid=18396, in which Rules Number 31 and 32 are Acid-Base and Strong Acid-Base.

If all pieces of both sides changes, why not make transformation optional?
Another idea: when player makes move, he choses to change either kind of movement (rook, bishop, knight, camel, etc.) or capturing method (replacement, overtaking, custodianship, etc.)? 'Pieces of same type', wich also changes, are only pieces, wich have both same kind of movement and same capturing methods. Or ALL pieces, wich had property,  wich was changed (all pieces, moving as rook or all pieces, capturing byovertaking) also transforms! Or, for example, if piece, wich had knight's move changed capturing method into custodianship, all piece with knight's move's now captures by custodianship.
Capturing with special methods by knight must be specified: knight's leap can be considered as one step knightrider or as another way of movement. In second case, knight's move can be defined as moving 2 steps orthogonally and one at right angles (and camel's move in this case is '3 orthogonally and 1 at right angles' or '2 diagonally and 1 at right angles') or as moving 1 orthogonally and 1 diagonally outwards.

How about combining Polypiece changes with board changes at the same time,
http://www.chessvariants.org/index/msdisplay.php?itemid=MScentralrotatio? Or Betza's own Closing Time, http://www.chessvariants.org/boardrules.dir/closingtime.html, which could mean in as one piece and out as another type. And does not Big Outer(year 1999) have some of the features before Polypiece(2003), or certainly before the above combination of both piece-type and board change suggested by joining Betza's Polypiece and Gifford's Central Rotation?  In Blanchard's Big Outer it is change of movement by board position: 
http://www.chessvariants.org/large.dir/contest/bigouter.html.

What was happening 10 years ago?  When a piece moves, it not only changes its type but does so for all the pieces of its type both sides.  The inspiration was Aronson's Chess With Cyclical Armies, which changes only one unit at a time.  The follow-up was Fourriere's Pocket Polypiece, which uses a Nelson Pocket Mutation pocket -- Pocket Mutation being the most played nontraditional chess at Game Courier.  All CVs in the polypiece genre have questionable clarity, and Pocket Polypiece is the most extreme.  

Clearer than the others, Pocket Mutation itself has very moderate Betzan polypiece quality because each change is spread out over two moves a piece at a time.

''Clarity'' goes back to mathematician Thompson's ''Defining the Abstract,'' which sees  constant trade-off in game design pairwise Drama versus Decisiveness and Depth versus Clarity. 
Http://www.thegamesjournal.com/articles/DefiningtheAbstract.shtml.

Any Ralph Betza polypiece chess is going to be deep but may lack clarity. This particular article by Betza is suggestive of thousands of different CVs having polypieces.