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There's some contention in my group about whether all three units consolidate or just the unit/model that dealt the killing blow.

It's in the FAQ. All units that Charge must be selected to fight before the Fight phase ends. Any unit that is selected to fight may Pile In and Consolidate.

BRB FAQ

Q: If a unit starts the Fight phase within 1" of an enemy unit
but, due to casualties suffered and models removed, it is no
longer within 1" of any enemy units later in the phase, can it
still be chosen to fight?
A: No, unless the unit in question charged this turn it
cannot be chosen to fight if there are no longer enemy
units within 1". Note that it is possible – if heavy enough
casualties have been incurred – that even a unit that
did charge may not be close enough, even after a pile in move, to make close combat attacks. It can still, of
course, consolidate.

A unit that charges will always pile in and consolidate, even if it can't make any attacks.

*may always pile in and consolidate. Its always optional, and even if you choose to do it you can do it 0".

 BlaxicanX wrote:
There's some contention in my group about whether all three units consolidate or just the unit/model that dealt the killing blow.

The procedure would be as follows:

Units A, B, and C charge
Unit A piles in
Unit A fights
Unit A consolidates
Unit B piles in
Unit B fights
Unit B consolidates
Unit C piles in
Unit C fights
Unit C consolidates

Piling in and consolidating are part of the unit's activation on the fight phase, so the first activated unit will completely finish consolidating in accordance with the rules for doing so before the next activated unit gets to being piling in.

So the answer to the letter of your question is yes. Unless otherwise forbidden, all units that charge get to fight, pile in, and consolidate even if they have no valid targets to fight.
However, the answer to the spirit of your question is "probably no", only the unit that scored the killing blow and any that follow it consolidate meaningfully, because units that didn't destroy it will generally have to be consolidating into the still-yet-undestroyed unit or remaining stationary.