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How does the throat size and back bore of a mouthpiece affect the playing? Is it in articulation, response, or a darker vs brighter tone?

Yes, I believe so!

Any specifics on how the sizes affect things? I'm still trying to refine my mouthpiece hunt.

All the above. Bach goes into great detail in their mouthpiece catalog about bore and throats, including, at least for their trumpet mouthpieces, a description of what each backbore does in all these regards.

Well, in addition to the factors you mention, there's also said to be some effect here on "slotting", the range of pitches that can easily be produced with a particular fingering.

short version:

as backbore gets bigger, "popping pitch" gets higher
as cup volume gets bigger, "popping pitch" gets lower

so, to first order, within reason, <insert weasel words here>, if you increase BOTH the
backbore AND the cup volume (by changing depth or width, say), then the "popping pitch" stays
more or less constant.

But...to SECOND order...

as backbore gets bigger, the WIDTH of "slots" gets bigger, but the HEIGHT of the "slots" gets smaller.

Which means: small bore is good for beginners, who need tight slots (because they buzz white noise and depend on the horn to select the pitch). large bore is preferred by those with pro chops, because those folk are buzzing the right pitch and want the freedom to lip the pitch up or down.

There are many simplifications in the above, and probably lots of errors - but it's a model that
does a pretty good job of prediction.

The problem for the buyer is: you can't just look at ONE of those parameters - you really need
to consider BOTH throat diameter AND cup volume.

http://cderksen.home.xs4all.nl/images/C ... ooklet.pdf

Here is a link which shows how certain volumes, back bores and throat sizes can affect intonation.

I would not argue with Bach, Schilke, or Warburton about the backbore of trumpet mouthpieces, but I would point out that scaling up a trumpet backbore to tuba size would make it four times longer. So it seems that trumpet backbore corresponds mostly to the leadpipe of a tuba.

My two cents would be this: If you need to work on cleaning up your pianissimo, choose a small throat. If you just want to get a big sound, choose a larger throat.

A few years ago I had Doug Elliot make me a shank that was one inch longer than standard, which made one of my tubas more comfortable to play, and allowed me to push my tuning slide in about half an inch. That was a fairly significant change in the backbore taper, but I did not notice any effect on my tone, intonation, or response.

Some invaluable info at Stork's website:
http://storkcustom.com/Library.htm" target="_blank