Fuel flow and leaning this time

Yet another question.

Say you're at altitude and it's time to lean the constant speed engine(s).

You pull back on the mixture control, the EGT starts to go up. The engine
"sounds" more powerful as you do this. The prop might speed up momentarily
until the governor does its stuff and keeps it at the selected speed. You
reach peak EGT, and if for argument's sake you're going for a certain
degrees ROP, you ease the mixture forward until you hit the target temp.

While you're doing this, you're also watching the fuel flow meter that you
have installed. As you pull the mixture back, the fuel flow starts to go up
as the power increases, and peaks with peak EGT. After peak the fuel flow
would go down as you keep on pulling mixture back.

The thing that has me a bit confused is this:-

I understand that roughly speaking increased power = increased fuel flow.
Therefore, it makes theoretical sense to me that as you pull the mixture
back and get closer to stoichiometric that you will see an increase in fuel
flow because the power output of the engine is increasing. What I can't do
is explain why mechanically.

If you've got a constant speed engine that is running full rich and is at a
certain RPM and MP setting, when you pull the mixture back, it'll still be
running at the same MP and RPM, right (after the governor has done its
thing)? Therefore, the pistons will be hopping up and down at the same rate,
won't they? If so, why is more fuel getting used when leaned as opposed to
full rich? For some reason I seem to think that it should be *less* fuel
flow but burned more efficiently, but that obviously isn't the case,
demonstrated empirically by the fact that fuel flow increases, and also due
to the maxim that more power requires more fuel.

I realise the answer must be pretty ~duh~ obvious, but it's escaping me at
the moment.

Once again, thanks in advance!